Relations between crossed modules of different algebras

Relations between crossed modules of different algebras RAFAEL FERN ´ ANDEZ CASADO 2015 Relations between crossed modules of different algebras by RAFAEL FERN ´ ANDEZ CASADO DISSERTATION Submitted for the degree of DOCTOR EN MATEM ´ ATICAS en la UNIVERSIDAD DE SANTIAGO DE COMPOSTELA Santiago de Compostela, 2015 Relations between crossed modules of different algebras Fdo.: Rafael Fern´andez Casado Memoria para optar al grado de Doctor realizada en el Departamento de ´ Algebra de la Universidad de Santiago de Compostela bajo la direcci´on de los Profesores D. Emzar Khmaladze y D. Manuel Ladra Gonz´alez. Santiago de Compostela, a 28 de septiembre de 2015. Fdo.: Emzar Khmaladze Fdo.: Manuel Ladra Gonz´alez Relations between crossed modules of different algebras Dr. Emzar Khmaladze y Dr. Manuel Ladra Gonz´alez, AUTORIZAMOS la presentaci´on de la Tesis Doctoral con t´ıtulo Relations between crossed modules of different algebras, realizada por D. Rafael Fern´andez Casado bajo nuestra direcci´on en el Departamento de ´ Algebra de la Universidad de Santiago de Compostela, para optar al grado de Doctor por la Universidad de Santiago de Compostela. Santiago de Compostela, a 28 de septiembre de 2015. Fdo.: Emzar Khmaladze Fdo.: Manuel Ladra Gonz´alez Resumen de la Tesis Doctoral: Relaciones entre m´odulos cruzados de diferentes ´algebras Resumen abreviado: En el presente trabajo extendemos a m´odulos cruzados la adjunci´on entre el funtor liezaci´on LieAs :As →Lie y el funtor ´algebra envolvente universal U:Lie →As. Adem´as, probamos que existe un isomorfismo entre las categor´ıas de m´odulos por la izquierda sobre un m´odulo cruzado de ´algebras de Lie y m´odulos por la izquierda sobre su m´odulo cruzado envolvente universal. Asimismo, construimos una generalizaci´on a dimensi´on 2 de la adjunci´on entre el funtor Lb :Dias →Lb y el funtor di´algebra envolvente universal Ud:Lb →Dias. Debido a que esta ´ultima generalizaci´on involucra a los m´odulos cruzados de di´algebras, damos una definici´on adecuada de los mismos, basada en la definici´on general de m´odulos cruzados en categor´ıas de inter´es. Adem´as, definimos el concepto de 2-di´algebra estricta, por analog´ıa con la noci´on de 2-´algebra asociativa estricta. Asimismo, probamos que las categor´ıas de m´odulos cruzados de di´algebras y 2-di´algebras estrictas son equivalentes. Tambi´en construimos la di´algebra de los tetramultiplicadores, que resultar´a ser el actor en la categor´ıa de di´algebras bajo ciertas condiciones. Adem´as, a partir de un m´odulo cruzado de ´algebras de Leibniz, construimos un actor general para el mismo, que resultar´a ser el actor en ciertos casos particulares. El concepto de m´odulo cruzado de grupos fue formulado por primera vez por Whitehead a finales de la d´ecada de los 40 [83]. Poco despu´es, Mac Lane y Whitehead [70] probaron que los m´odulos cruzados pueden utilizarse como modelo algebraico para los CW-espacios conexos cuyos grupos de homotop´ıa son triviales en dimensi´on mayor que 2. Los m´odulos cruzados generalizan al mismo tiempo los conceptos de subgrupo normal y m´odulo sobre un grupo. Desde su introducci´on han jugado un papel muy importante en diversas ´areas de las matem´aticas, en particular en teor´ıa de homotop´ıa. Por ejemplo, aparecen en varios problemas de clasificaci´on de tipos homot´opicos en dimensi´on baja y en las generalizaciones del Teorema de van Kampen. M´as all´a de su valor como herramienta para la teor´ıa de homotop´ıa, los m´odulos cruzados han sido estudiados como objetos algebraicos de propio derecho. Por ejemplo, Norrie extendi´o a m´odulos cruzados algunos conceptos y estructuras propias de la teor´ıa de grupos en su tesis doctoral [72]. En particular, construy´o el actor de un m´odulo cruzado de grupos e introdujo la noci´on de centro de un m´odulo cruzado, as´ı como los conceptos de m´odulo cruzado completo y perfecto. ix y las acciones de pyqsobre nson compatibles, es decir, [n, [p, q]] = [[n, p], q]−[[n, q], p], [p, [n, q]] = [[p, n], q]−[[p, q], n], [p, [q, n]] = [[p, q], n]−[[p, n], q], [n, [q, p]] = [[n, q], p]−[[n, p], q], [q, [n, p]] = [[q, n], p]−[[q, p], n], [q, [p, n]] = [[q, p], n]−[[q, n], p], para todo n∈n,p∈pyq∈q. (ii) Existen dos aplicaciones K-bilineales ξ1:m×q→nyξ2:q×m→ntales que µξ2(q, m)=[q, m], µξ1(m, q)=[m, q], ξ2(µ(n), m)=[n, m], ξ1(m, µ(n)) = [m, n], ξ2(q, [p, m]) = ξ2([q, p], m)−[ξ2(q, m), p], ξ1([p, m], q) = ξ2([p, q], m)−[p, ξ2(q, m)], ξ2(q, [m, p]) = [ξ2(q, m), p]−ξ2([q, p], m), ξ1([m, p], q) = [ξ1(m, q), p]−ξ1(m, [q, p]), ξ2(q, [m, m0]) = [ξ2(q, m), m0]−[ξ2(q, m0), m], ξ1([m, m0], q) = [ξ1(m, q), m0]−[m, ξ2(q, m0)], ξ2([q, q0], m) = [ξ2(q, m), q0]+[q, ξ2(q0, m)], ξ1(m, [q, q0]) = [ξ1(m, q), q0]−[ξ1(m, q0), q], [q, ξ1(m, q0)] = −[q, ξ2(q0, m)], ξ1(m, [p, q]) = −ξ1(m, [q, p]), [p, ξ1(m, q)] = −[p, ξ2(q, m)], para todo m, m0∈m,n∈n,p∈p,q, q0∈q. Adem´as, si al menos una de las siguientes condiciones se cumple, el enunciado rec´ıproco tambi´en es cierto. Ann(n) = 0 = Ann(q), Ann(n)=0 y[q,q] = q, [n,n] = ny[q,q] = q. Para poder confirmar que la colecci´on de ecuaciones del teorema anterior define un conjunto de acciones derivadas de un m´odulo cruzado de ´algebras de Leibniz sobre xvi otro, tenemos que comprobar que a partir de dichas ecuaciones podemos definir el producto semidirecto de los dos m´odulos cruzados correspondientes. Dados (m,p, η) y (n,q, µ) dos m´odulos cruzados de ´algebras de Leibniz tales que las condiciones (i) y (ii) del Teorema 2.2.18 se cumplen, al existir acciones de msobre n y de psobre q, es posible considerar los productos semidirectos de ´algebras de Leibniz nomyqop. Adem´as, tenemos el siguiente resultado. Teorema 2.2.21. Existe una acci´on del ´algebra de Leibniz qopsobre el ´algebra de Leibniz nom, dada por [(q, p),(n, m)] = ([q, n]+[p, n] + ξ2(q, m),[p, m]), [(n, m),(q, p)] = ([n, q]+[n, p] + ξ1(m, q),[m, p]), para todo (q, p)∈qop,(n, m)∈nom, con ξ1yξ2definidas como en el Teorema 2.2.18. Adem´as, el morfismo de ´algebras de Leibniz (µ, η): nom→qop, dado por (µ, η)(n, m)=(µ(n), η(m)), para todo (n, m)∈nomes un m´odulo cruzado de ´algebras de Leibniz junto con la acci´on anterior. Este ´ultimo resultado nos permite definir el producto semidirecto de m´odulos cruzados de ´algebras de Leibniz (n,q, µ) y (m,p, η) como el m´odulo cruzado (no m,qop,(µ, η)). Adem´as, estamos en condiciones de escribir la siguiente definici´on: Definici´on 2.2.23. Si (m,p, η)y(n,q, µ)son dos m´odulos cruzados de ´algebras de Leibniz y se verifica al menos una de las siguientes condiciones, 1. Ann(n) = 0 = Ann(q), 2. Ann(n) = 0 y[q,q] = q, 3. [n,n] = ny[q,q] = q, entonces una acci´on del m´odulo cruzado (m,p, η)sobre (n,q, µ)es un morfismo de m´odulos cruzados de ´algebras de Leibniz de (m,p, η)en Act(n,q, µ). En otras palabras, bajo una de esas tres condiciones, Act(n,q, µ)es el actor de (n,q, µ). Los pasos en la construcci´on de (Bider(q,n),Bider(n,q, µ),∆) sugieren candidatos claros para la extensi´on a m´odulos cruzados del ´algebra de los bimultiplicadores y de la di´algebra de los tetramultiplicadores. Estas generalizaciones ser´an consideradas en futuros trabajos. Cerramos el segundo cap´ıtulo con las definiciones de m´odulos por la izquierda sobre un m´odulo cruzado de ´algebras de Lie y de ´algebras asociativas, ya que una de las consecuencias de uno de nuestros resultados principales, que aparece en el ´ultimo cap´ıtulo, ser´a el isomorfismo entre las categor´ıas de m´odulos por la izquierda sobre xvii un m´odulo cruzado de ´algebras de Lie y m´odulos por la izquierda sobre su m´odulo cruzado envolvente universal. En el reciente art´ıculo [22], los autores construyen un par de funtores adjuntos entre las categor´ıas de m´odulos cruzados de grupos y ´algebras asociativas unitarias. Estos funtores son una generalizaci´on natural de la adjunci´on cl´asica entre el funtor grupo de las unidades y el funtor ´algebra de grupo. Esta ´ultima adjunci´on tiene un an´alogo para las categor´ıas de ´algebras de Lie y ´algebras asociativas, dado por la adjunci´on entre el funtor liezaci´on, que le da a cada ´algebra Aestructura de ´algebra de Lie mediante el corchete [a, b] = ab −ba,a, b ∈A, y el funtor que asigna a cada ´algebra de Lie psu ´algebra envolvente universal U(p). Adem´as, existe una adjunci´on entre el funtor que asigna a cada di´algebra Del corchete de Leibniz dado por [x, y] = xay−y`xpara todo x, y ∈D, y el funtor di´algebra envolvente universal (ver [65]). Comenzamos el ´ultimo cap´ıtulo recordando la construcci´on de la extensi´on a m´odulos cruzados de la adjunci´on entre el funtor grupo de las unidades y el funtor ´algebra de grupo. Adem´as, en la Subsecci´on 3.1.2, demostramos que la generalizaci´on a m´odulos cruzados del segundo de esos funtores no tiene un comportamiento natural con su versi´on en dimensi´on 1, en el sentido de que el siguiente diagrama de categor´ıas y funtores, donde E1(G) = (G, G, idG) y E0 1(A) = (A, A, idA), Gr XGr As1XAs1. E1 K XK E0 1 no es conmutativo, ni siquiera salvo isomorfismo. En las Secciones 3.2 y 3.3 presentamos la generalizaci´on a m´odulos cruzados de las adjunciones entre las categor´ıas Lie vs As yLb vs Dias mencionadas anteriormente, cumpliendo as´ı uno de los objetivos principales de este trabajo. En ambos casos, primero construimos las correspondientes extensiones a m´odulos cruzados y despu´es comprobamos el buen comportamiento de las mismas a trav´es de los siguientes resultados. Es importante tener en cuenta que en la generalizaci´on de los funtores ´algebra envolvente universal y di´algebra envolvente universal, los cat1-objetos juegan un papel fundamental. Para XLie vs XAs: Teorema 3.2.5. El funtor XU es adjunto por la izquierda del funtor XLieAs. xviii Teorema 3.2.6. Los cuadrados interiores y exteriores de los siguientes diagramas son conmutativos o conmutan salvo isomorfismo para i= 0,1. As Lie As Lie XAs XLie XAs XLie ⊥ LieAs I0 i a U Ii` ⊥ LieAs I0 ia U Ii ` > XLieAs Φ0 i XU Φi > XLieAs Φ0 i+1 XU Φi+1 Adem´as, tenemos el siguiente resultado: Teorema 3.2.8. Sea (m,p, ν)un m´odulo cruzado de ´algebras de Lie. Entonces, las categor´ıas de (m,p, ν)-m´odulos por la izquierda y XU(m,p, ν)-m´odulos por la izquierda son isomorfas. Para XLb vs XDias: Teorema 3.3.4. El funtor XUdes adjunto por la izquierda del funtor XLb. Teorema 3.3.5. Los cuadrados interiores y exteriores de los siguientes diagramas son conmutativos o conmutan salvo isomorfismo para i= 0,1. Dias Lb Dias Lb XDias XLb XDias XLb ⊥ Lb J0 i a Ud Ji` ⊥ Lb J0 ia Ud Ji ` > XLb Ψ0 i XUd Ψi > XLb Ψ0 i+1 XUd Ψi+1 Finalmente, en la ´ultima secci´on del ´ultimo cap´ıtulo, completamos los siguientes diagramas, formados por cuadrados interiores y exteriores conmutativos (o que conmutan salvo isomorfismo) para i= 0,1: As Dias As Dias XAs XDias XAs XDias ⊥ ⊂ I0 i a As J0 i` ⊥ ⊂ I0 ia As J0 i ` > ⊂ Φ0 i XAs Ψ0 i > ⊂ Φ0 i+1 XAs Ψ0 i+1 xix Lie Lb Lie Lb XLie XLb XLie XLb ⊥ ⊂ Ii a LieLb Ji` ⊥ ⊂ Iia LieLb Ji ` > ⊂ Φi XLieLb Ψi > ⊂ Φi+1 XLieLb Ψi+1 XAs XLie XDias XLb ⊥ XLieAs ⊂ a XU ⊂` > XLb XAs XUd XLieLb Estos diagramas, junto al ya conocido As Lie Dias Lb ⊥ LieAs ⊂ a U ⊂` > Lb As Ud LieLb nos permiten construir los cuatro paralelep´ıpedos protagonistas del teorema que cierra la tesis: xx Teorema 3.4.3. En los siguientes paralelep´ıpedos de categor´ıas y funtores As Lie Dias Lb XAs XLie XDias XLb ⊥ LieAs I0 i a ⊂ ` U ` Ii ` ⊂ Lb > J0 i a As Ud Ji` LieLb Φ0 i ⊥ XLieAs ⊂ ` Φi XU ` ⊂ Ψ0 i XLb > XAs Ψi XUd XLieLb As Lie Dias Lb XAs XLie XDias XLb ⊥ LieAs I0 ia ⊂ ` U `Ii ` ⊂ Lb > J0 ia As Ud Ji ` LieLb Φ0 i+1 ⊥ XLieAs ⊂ ` Φi+1 XU ` ⊂ Ψ0 i+1 XLb > XAs Ψi+1 XUd XLieLb todos los cuadrados interiores y exteriores de funtores adjuntos son conmutativos o conmutan salvo isomorfismo para i= 0,1. Es importante tener en cuenta que en cada una de las caras de los paralelep´ıpedos, los adjuntos por la izquierda forman los cuadrados exteriores y los adjuntos por la derecha los cuadrados interiores. xxi xxii Acknowledgements Firstly, I would like to express my sincere gratitude to my advisors Prof. Manuel Ladra and Prof. Emzar Khmaladze for their guidance and tremendous support throughout the course of my research. There are no proper words to describe how grateful I am for their help, dedication, patience and unconditional human support. I would like to stress the close and familiar treatment provided by Prof. Manuel Ladra, which was essential for me to find the required self-confidence to finish this work. His backing in those moments of hesitation and slow progress has been more important than any other mathematical advice. I am also indebted to Prof. Jos´e Manuel Casas Mir´as for his help in proofreading the subsection on the actor crossed module of Leibniz algebras and his words of encouragement. Thanks are also due to Prof. Elena V´azquez Abal for her advice and unselfish support. I would also like to thank my family and friends, who helped me and understood me during this particularly intense and demanding moment of my life. I am especially grateful to Patri, who kindly cared for me when I needed it the most. xxiv Introduction Aims and background The concept of crossed module of groups was formulated for the first time by Whitehead in the late 1940s [83]. Soon after, Mac Lane and Whitehead [70] proved that crossed modules work as an algebraic model for path-connected CW-spaces whose homotopy groups are trivial in dimensions greater than 2. They are algebraic objects with a rich structure and provide a generalization of both the concepts of normal subgroup and module over a group. From the very beginning, crossed modules of groups have played an important role in several areas of mathematics, particularly in homotopy theory. For example, they appear in various classification problems for low-dimensional homotopy types and the derivation of van Kampen theorem generalizations (see the survey by Brown [12]). Beyond their value as a tool for homotopy theory, crossed modules have been studied as algebraic objects in their own right. For instance, Norrie generalized some group theoretic concepts and structures to crossed modules in her PhD thesis [72]. In particular, she defined the actor of a crossed module, as well as the notions of centre of a crossed module, complete and perfect crossed modules. The study of (co)homological properties of crossed modules of groups has been the subject of several papers. We point out two (co)homology theories of crossed modules of groups, one introduced and investigated in the works of Baues [5] and Ellis [39] via classifying spaces, and the other defined by Carrasco, Cegarra and R.-Grandje´an [15] as cotriple (co)homology. Later, R.-Grandje´an, Ladra and Pirashvili [49] found a relation between these two homology theories. Crossed modules of different algebraic objects, not only groups, have also been studied. For instance, in [60] Kassel and Loday used Lie crossed modules as computational tools in order to give an interpretation of the third relative Chevalley-Eilenberg cohomology of Lie algebras. Internal (cotriple) homology and Chevalley-Eilenberg homology theories of Lie crossed modules were investigated in [23, 35]. Lie crossed modules also occur in the “categorification” problem of the theory of Lie algebras [4] as an equivalent formulation of strict Lie 2-algebras. From the 1960s, many authors have attempted to answer the question of what xxv 3.2.3 Adjunction between XLie and XAs ....................112 3.3 XLb vs XDias ....................................116 3.3.1 From XDias to XLb ............................117 3.3.2 Universal enveloping crossed module of a Leibniz crossed module . . . 119 3.3.3 Adjunction between XLb and XDias ...................122 3.4 Relations between crossed modules of Lie, Leibniz, associative algebras and dialgebras......................................126 3.4.1 Adjunction between XAs and XDias ...................127 3.4.2 Adjunction between XLie and XLb ....................128 3.4.3 Extended diagram for categories of crossed modules . . . . . . . . . . 129 4 Conclusions and further research 133 Bibliography 135 xxxii Chapter 1 Crossed modules and equivalent structures In Section 1.1 we recall the well-known equivalence between crossed modules and internal categories in a category of groups with operations. We gather some indispensable notions stated by Orzech [74] and the equivalence itself, proved by Porter [78]. In Section 1.2 we sketch the well-known equivalence between cat1-objects and internal categories in a category of groups with operations. In the five contained subsections we give some basic definitions and explore essential properties of crossed modules in the five particular categories considered in this thesis. Furthermore we give an explicit description of the equivalence between crossed modules and cat1-objects for the five different situations, since the aforementioned equivalence is essential for a proper comprehension of the proofs in Chapter 3. All the results were known prior to this thesis, although crossed modules of dialgebras had not been explicitly described so far. Finally, in Section 1.3 we introduce the notion of strict 2-dialgebra by analogy to the concept of strict associative 2-algebra by Khmaladze [61] and the one of strict Lie 2-algebra by Baez [4]. Additionally, we prove that strict 2-dialgebras and strict Leibniz 2-algebras are equivalent to crossed modules of dialgebras and Leibniz algebras respectively. 1.1 Crossed modules and internal categories 1.1.1 Crossed modules of Ω-groups The next definition can be found in [19, 20, 71, 74, 78] (for additive notation) and [76] (for multiplicative notation). It is based on the more general notion of category 1 2 1 Crossed modules and equivalent structures of groups with multiple operators introduced by Higgins [52]. Definition 1.1.1. A category of groups with operations (or Ω-groups) is a category Cwhose objects are groups with a set of operations Ωand with a set of identities E, such that Eincludes the group laws and the following conditions hold. If Ωiis the set of i-ary operations in Ω, then: (a) Ω=Ω0∪Ω1∪Ω2. (b) The group operations (written using additive notation: 0,−,+) are elements of Ω0,Ω1and Ω2respectively. Let Ω0 1= Ω1\ {−} and Ω0 2= Ω2\ {+}. If ∗ ∈ Ω0 2, then ∗◦∈Ω0 2, with x1∗◦x2=x2∗x1. Besides, Ω0={0}. (c) For any ∗ ∈ Ω0 2,Econtains the identity x1∗(x2+x3)=(x1∗x2)+(x1∗x3). (d) For any ω∈Ω0 1and ∗ ∈ Ω0 2,Econtains the identities: ω(x1+x2) = ω(x1) + ω(x2), ω(x1∗x2) = ω(x1)∗x2. A morphism of Ω-groups is a set map which preserves all the operations. Remark 1.1.2. It is important to note that the group operation is not necessarily commutative, hence −(x1+x2) = −x2−x1for all x1, x2∈C, with Can object in category of groups with operations, in contrast to the first identity from (d). Besides, the fact that ∗◦∈Ω0 2for any ∗ ∈ Ω0 2will allow us to disregard the right sided version of many identities involving operations in Ω0 2, as they will follow immediately from the left sided version. The following lemma is an immediate consequence of the group structure and the axiom (c) from Definition 1.1.1. Lemma 1.1.3. Let Cbe a category of groups with operations and Can object in C. Then, (i) x∗0=0, (ii) −(x1∗x2) = −x1∗x2, (iii) x1∗x2+x3∗x4=x3∗x4+x1∗x2, for all x, x1, x2, x3, x4∈C,∗ ∈ Ω0 2. In [74], Orzech introduced the notion of category of interest, which is no more than a category of groups with operations that verifies some extra conditions. Definition 1.1.4. A category of interest is a category of groups with operations which satisfies two additional axioms: 1.1.1 Crossed modules of Ω-groups 3 (1) x1+ (x2∗x3)=(x2∗x3) + x1. (2) For any ordered pair (∗,¯ ∗)∈Ω0 2×Ω0 2, there is a word Wsuch that, (x1∗x2)¯ ∗x3=W(x1(x2x3), x1(x3x2),(x2x3)x1,(x3x2)x1, x2(x1x3), x2(x3x1),(x1x3)x2,(x3x1)x2), where each juxtaposition represents an operation in Ω0 2. The reason for us to include the previous definition is that our principal results establish relations between crossed modules in categories that satisfy the foregoing axioms. Categories of groups Gr, Lie algebras Lie, associative algebras As, Leibniz algebras Lb and associative dialgebras Dias can be found among the examples of categories of interest provided in [71], along with the counterexample of Jordan algebras, for which axiom (2) fails. Let Cbe a category of groups with operations. It is possible to define the notions of action and semidirect product in such a category. Definition 1.1.5 ([74]).Let Aand Bbe objects in C. An extension of Bby Ais a sequence 0A C B 0 ι σ in which σis surjective and ιis the kernel of σ. We say that an extension is split if there is a morphism ε:B→Csuch that σε = idB. A split extension of Bby A is called B-structure on A. An extension is singular if Ais singular, that is, if Ais abelian as a group and a1∗a2= 0 for all a1, a2∈A,∗ ∈ Ω0 2. Given a B-structure on A, there is an induced set of actions of Bon A, one for each operation in Ω2. If we assume A⊂C, with ιthe inclusion, the definition of those actions is as follows: ba=ε(b) + a−ε(b), b∗a=ε(b)∗a, for all a∈A,b∈B. Actions arising from split extensions are called derived actions in [74], where Orzech proves the following result: Theorem 1.1.6 ([74]).Let Aand Bbe objects in C. Given a set of actions of B on A(one for each operation in Ω2), the semidirect product AoB, which consists of A×Bas a set, with the operations: ω(a, b)=(ω(a), ω(b)),(1.1.1) (a, b)+(a0, b0)=(a+ba0, b +b0),(1.1.2) (a, b)∗(a0, b0)=(a∗a0+b∗a0+b0∗◦a, b ∗b0),(1.1.3) for all a, a0∈A,b, b0∈B,ω∈Ω0 1,∗ ∈ Ω0 2, is an object in Cif and only if the set of actions of Bon Ais a set of derived actions. 4 1 Crossed modules and equivalent structures Remark 1.1.7. Observe that −(a, b)=(−b(−a),−b), immediately from the definition of the addition in AoBtogether with the identities (1), (2) and (3) from the next lemma. In the categories of groups, Lie algebras, associative algebras, Leibniz algebras and associative dialgebras, derived actions will be called simply actions. However, in Chapter 2 we will use again the adjective “derived” whenever we need to stress that a set of actions is indeed induced by a split extension. In any case, the context will always make the difference clear. Due to the way a set of derived actions is defined, one can easily check that the identities in the following lemma hold. Lemma 1.1.8. Let Aand Bbe objects in C, together with a B-structure on A. Then: (1) 0a=a, (2) b(a1+a2) = ba1+ba2, (3) (b1+b2)a=b1(b2a), (4) b∗(a1+a2) = b∗a1+b∗a2, (5) (b1+b2)∗a=b1∗a+b2∗a, (6) (b1∗b2)(a1∗a2) = a1∗a2, (7) (b1∗b2)(b∗a) = b∗a, (8) a1∗(ba2) = a1∗a2, (9) b1∗(b2a) = b1∗a, (10) ω(ba) = ω(b)ω(a), (11) ω(b∗a) = ω(b)∗a=b∗ω(a), (12) x1∗x2+x3∗x4=x3∗x4+x1∗x2, for all a, a1, a2∈A,b, b1, b2∈B,x1, x2, x3, x4∈A∪B,ω∈Ω0 1,∗ ∈ Ω0 2. Proof. All the equalities can be easily proved by using the definition of the derived actions along with axioms (b), (c), (d) from Definition 1.1.1, Lemma 1.1.3 (ii) and the fact that ε(as in the definition of a B-structure) is a morphism. Remark 1.1.9. In [31], the previous equalities are proved to be not only necessary conditions but also sufficient to define a set of derived actions in a category of Ωgroups. That is the reason why we include all the equalities, although we will only make use of a few of them. Nevertheless, in the next section, for every specific category, we will define derived actions in terms of equations by using the description given in [20, p. 91] for the particular case of categories of interest, in which conditions (6) and (7) are replaced by b(a1∗a2) = a1∗a2, b1(b2∗a) = b2∗a, (b1∗b2)a=a, 1.1.1 Crossed modules of Ω-groups 5 and condition (12) is replaced by a1+ (b∗a2)=(b∗a2) + a1(as in axiom (1) from Definition 1.1.4) for all a, a1, a2∈A,b, b1, b2∈B,∗ ∈ Ω0 2. Besides, an additional condition is required, described as axiom (2) from Definition 1.1.4, but with x1, x2, x3∈A∪B. Now we can define crossed modules in terms of actions and operations. Definition 1.1.10. A crossed module in a category of groups with operations Cis a triple (A, B, µ), where µis a morphism between the objects Aand B, together with a B-structure on A, such that for all a, a1, a2∈A,b∈B,∗ ∈ Ω0 2, µ(ba) = b+µ(a)−b, µ(b∗a) = b∗µ(a),(CM1) µ(a1)a2=a1+a2−a1, µ(a1)∗a2=a1∗a2.(CM2) Note that Porter [78] includes the identities µ(a∗b) = µ(a)∗bin (CM1) and a1∗µ(a2) = a1∗a2in (CM2), but those follow from µ(b∗a) = b∗µ(a) and µ(a1)∗a2= a1∗a2respectively, due to condition (b) in Definition 1.1.1 along with the way of defining a set of derived actions from a B-structure. Remark 1.1.11. The second axiom is usually called Peiffer identity, while the first one is sometimes referred to as equivariance. Both (CM1) and (CM2) will have specific descriptions and tags for the five particular categories considered (see Subsections 1.2.1–1.2.5). However, we will sometimes write simply equivariance or Peiffer identity since the context itself will clarify the category in use. A precrossed module is a triple (A, B, µ), together with a B-structure on A, that only satisfies the equivariance condition. Definition 1.1.12. Given two crossed modules (A, B, µ)and (A0, B0, µ0), a morphism of crossed modules is a pair (ϕ, ψ)of morphisms in C,ϕ:A→A0and ψ:B→B0, such that µ0ϕ=ψµ and: ϕ(ba) = ψ(b)ϕ(a),(1.1.4) ϕ(b∗a) = ψ(b)∗ϕ(a),(1.1.5) for all a∈A,b∈B. Composition of morphisms of crossed modules is defined component-wise and the identity morphism is given by (idA,idB) for any crossed module (A, B, µ). We will denote by XMod(C) the category of crossed modules and morphisms of crossed modules in C. However, in Section 1.2 we will present specific notation for the five particular categories considered in this thesis, together with several examples and essential properties. 6 1 Crossed modules and equivalent structures 1.1.2 Internal categories Internal categories were introduced by Ehresmann [40, 41] although a more accessible description can be found in [3, 4, 9, 44, 76]. Categories within a category can be defined in a more general context than that of Ω-groups, so let us assume for the rest of this subsection that Cis simply a category with pullbacks. Definition 1.1.13. An internal category Cin Cconsists of an object of objects, C0, an object of arrows, C1, and the diagram: C1×C0C1C1C0 κ t s e where s, t are the source and target maps, eis the identity-assigning map, κis the composition map, C1×C0C1is the pullback: C1×C0C1C1 C1C0 π1 π2 s t (1.1.6) and the following diagrams commute, expressing the usual category laws: C0C1C0C1 C0C0 idC0 e sidC0 e t(1.1.7) C1×C0C1C1C1×C0C1C1 C1C0C1C0 π1 κs π2 κt st (1.1.8) C1×C0C1×C0C1C1×C0C1 C1×C0C1C1 κ×C0idC1 idC1×C0κκ κ (1.1.9) C0×C0C1C1×C0C1C1×C0C0 C1 e×C0idC1 π2 κ idC1×C0e π1 (1.1.10) 1.1.2 Internal categories 7 Note that C0×C0C1and C1×C0C0are respectively the pullbacks C0×C0C1C1 C0C0 π1 π2 s idC0 and C1×C0C0C0 C1C0. π1 π2 idC0 t Whenever we want to consider an internal category, we will simply write a sextuple C= (C1, C0, s, t, e, κ). Furthermore, since objects have elements in the categories we will use, those in C1will be denoted by letters typically used for morphisms in order to establish a complete analogy with the classical notion of category. Note that if one thinks of the elements in C1as maps, the disposition of π1and π2in diagrams (1.1.8) implies that κ(f, g) = g◦fin standard notation. Definition 1.1.14. Let C= (C1, C0, s, t, e, κ)and C0= (C0 1, C0 0, s0, t0, e0, κ0)be two internal categories. An internal functor F:C→C0consists of a pair (F1, F0)of morphisms in C,F1:C1→C0 1,F0:C0→C0 0, such that the following diagrams commute: C1C0C1C0C1C0 C0 1C0 0C0 1C0 0C0 1C0 0 s F1F0 t F1F0F1 e F0 s0t0e0 (1.1.11) C1×C0C1C0 1×C0 0C0 1 C1C1 F1×F0F1 κκ0 F1 (1.1.12) where F1×F0F1is given by: C1×C0C1C1 C0 1×C0 0C0 1C0 1 C1C0 1C0 0. π2 π1 F1×F0F1F1 π0 2 π0 1s0 F1t0 (1.1.13) 8 1 Crossed modules and equivalent structures Composition of internal functors is defined in the obvious way. We will denote by ICat(C) the category of internal categories and internal functors in C. It is possible to introduce the notion of internal natural transformation between two internal functors in Cand prove that internal categories, functors and natural transformations form a 2-category (see [4, Section 2]). 1.1.3 Equivalence between crossed modules and internal categories The equivalence between ICat(C) and XMod(C) was proved by Porter in [78] for C a category of groups with operations. Note that this equivalence has been recently extended to the more general situation of semiabelian categories (see [58, 59]). Nevertheless, for the purposes of the present work, we will recall Porter’s construction. Let us assume for rest of this subsection that Cis a category of Ω-groups. Lemma 1.1.15. Let C= (C1, C0, s, t, e, κ)be an internal category in C. Then, C1×C0C1={(f, g)|f, g ∈C1, t(f) = s(g)}, with the operations given by: ω(f, g)=(ω(f), ω(g)), (f, g)∗(f0, g0)=(f∗f0, g ∗g0), for all (f, g),(f0, g0)∈C1×C0C1,ω∈Ω1,∗ ∈ Ω2. Besides, κ((f∗f0),(g∗g0)) = κ(f, g)∗κ(f0, g0), for all (f, g),(f0, g0)∈C1×C0C1,∗ ∈ Ω2. These last identities are called the interchange laws. Proof. It is straightforward to check that {(f, g)|f, g ∈C1, t(f) = s(g)}is an object in C. Besides, it is immediate to prove that it is the pullback (1.1.6) with the projections π1and π2defined in the obvious way. On the other hand, κ:C1×C0C1→C1is a morphism in C. Therefore, given (f, g),(f0, g0)∈C1×C0C1,∗ ∈ Ω2, κ(f∗f0, g ∗g0) = κ((f, g)∗(f0, g0)) = κ(f, g)∗κ(f0, g0). We will say that the elements in C1×C0C1are pairs of “composable arrows”. Furthermore, internal categories in a category of groups with operations “have all their arrows invertible” in the following sense: Definition 1.1.16. An internal groupoid is an internal category C= (C1, C0, s, t, e, κ) for which, given any f∈C1, there is f0∈C1such that κ(f, f0) = es(f)and κ(f0, f) = et(f). 1.1.3 Equivalence between crossed modules and internal categories 9 Theorem 1.1.17. Every internal category in a category of Ω-groups is an internal groupoid. Proof. Given f∈C1, we define f−1=et(f)−f+es(f). It is clear that (f, f−1)∈ C1×C0C1, since s(f−1) = s(et(f)−f+es(f)) = t(f)−s(f) + s(f) = t(f), directly from the fact that sis a morphism and the commutativity of the first diagram in (1.1.7). Analogously, by using the fact that tis a morphism and the commutativity of the second diagram in (1.1.7), we get that (f−1, f)∈C1×C0C1. It is evident that (f, et(f)),(es(f), f),(es(f), es(f)) ∈C1×C0C1, so we can write the following: κ(f, f−1) = κ(f−es(f) + es(f), et(f)−f+es(f)) = κ(f, et(f)) −κ(es(f), f) +κ(es(f), es(f)) = f−f+es(f) = es(f), due to the interchange laws and the commutativity of diagram (1.1.10). One can similarly prove that κ(f−1, f) = et(f). Directly from the interchange laws, we get the following lemma: Lemma 1.1.18. Let C= (C1, C0, s, t, e, κ)be an internal category in C. Then: (i) κ(f, g) = f−es(g) + g=g−es(g) + ffor all (f, g)∈C1×C0C1, (ii) f+g=g+ffor all f, g ∈C1such that t(f) = 0 = s(g), (iii) f∗g= 0 for all f, g ∈C1such that t(f) = 0 = s(g),∗ ∈ Ω0 2. Proof. Let (f, g)∈C1×C0C1, that is t(f) = s(g). It is clear that (f, es(g)), (et(f), es(g)), (et(f), g) are pairs of composable arrows. Then we can write the following: κ(f, g) = κ(f−et(f) + et(f), es(g)−es(g) + g) =κ(f, es(g)) −κ(et(f), es(g)) + κ(et(f), g) =f−es(g) + g, as a consequence of the interchange laws (see Lemma 1.1.15) and the commutativity of diagram (1.1.10). Similarly, we can write: κ(f, g) = κ(et(f)−et(f) + f, g −es(g) + es(g)) =κ(et(f), g)−κ(et(f), es(g)) + κ(f, es(g)) =g−es(g) + f, so (i) holds. Let us show that (ii) follows immediately from (i). If t(f) = 0 = s(g), it is clear that (f, g)∈C1×C0C1and es(g) = 0. Due to (i), κ(f, g) = f−es(g) + g=f+g, but also κ(f, g) = g−es(g) + f=g+f. Hence, f+g=g+f. 16 1 Crossed modules and equivalent structures such that se = idC0=te, [Ker s, Ker t]=0, Ker s∗Ker t= 0, for all ∗ ∈ Ω0 2, where [Ker s, Ker t]is the commutator of Ker sand Ker t. Then C= (C1, C0, s, t, e, κ)is an internal category in C, with κgiven by κ(f, g) = f−es(g) + g for all (f, g)∈C1×C0C1. A diagram satisfying the hypotheses of the previous theorem is usually called cat1-object in C, alternatively described in the same way but with C0a subobject of C1. One can easily derive from this result and Lemma (1.1.18) the well-known equivalence between ICat(C) and the category of cat1-objects in C. Observe that we have not given a definition for morphisms of cat1-objects, but it is fairly obvious. The cat1-object structure will be essential in the proofs of the main results in Chapter 3. Hence, although the equivalence with crossed modules holds in general for any category of groups with operations, we will show the equivalence for the five particular cases considered in this thesis. Although we will not mention it for every particular case, all the definitions in the subsequent subsections agree with their corresponding general version for categories of groups with operations. Observe that it is not our intention to present a thorough review on crossed modules, but to introduce basic notions and tools required for the main results in this thesis. 1.2.1 The case of groups Crossed modules of groups were first described by Whitehead [83] in the late 1940s, as algebraic models for path-connected CW-spaces whose homotopy groups are trivial in dimensions >2. Since their first appearance, crossed modules have become an important tool in different areas of mathematics such as homotopy theory, group (co)homology [16] or K-theory. Observe that we will use multiplicative notation for the group operation, in contrast to the notation used in Section 1.1. Let us first recall what an action of a group on another group means in terms of equations. Definition 1.2.2. An action of a group Gon a group His a map G×H→H (g, h)7→ gh, such that: (1) 1h=h, 1.2.1 The case of groups 17 (2) g0(gh) = (g0g)h, (3) g(hh0) = ghgh0, for all g, g0∈G,h, h0∈H. Given an action of a Gon Hit is possible to define the semidirect product HoG as the underlying set of H×Gequipped with the operation given by: (h, g) (h0, g0)=(hgh0, gg0) for every (h, g),(h0, g0)∈H×G. The identity element is (1,1) and for any (h, g)∈ HoG, its inverse is given by ((g−1)(h−1), g−1). Definition 1.2.3. A crossed module of groups (H, G, ∂)is a group homomorphism ∂:H→Gtogether with an action of Gon Hsuch that ∂(gh) = g∂(h)g−1,(XGr1) ∂(h)h0=hh0h−1.(XGr2) for all h, h0∈H,g∈G. The first axiom is sometimes called equivariance, while the second one is usually known as Peiffer identity. If (H, G, ∂)satisfies (XGr1) but not necessarily (XGr2), it is called precrossed module. Directly from the definition, we get the following well-known result: Lemma 1.2.4. Given a crossed module (H, G, ∂), (i) Ker ∂is a normal subgroup of Hand Im ∂is a normal subgroup of G. (ii) Ker ∂⊂Z(H), where Z(H)is the centre of H. Proof. (i) The kernel of a group homomorphism is always a normal subgroup. Regarding Im ∂, let g0∈Im ∂and g∈G. There is h∈Hsuch that ∂(h) = g0. As a consequence of equivariance, gg0g−1=g∂(h)g−1=∂(gh)∈Im ∂. (ii) Let h∈Ker ∂and h0∈H. Then, due to the Peiffer identity, h0=1h0= ∂(h)h0=hh0h−1. Therefore h0h=hh0and Ker ∂⊂Z(H). Example 1.2.5. We recall some generic examples, which can be found, for instance, in [45, 72]. Let Gbe a group. (i) Gacts on any normal subgroup NCGby conjugation and the inclusion i:N ,→G together with that action is a crossed module. {1}and Gare normal subgroups of G, so any group Gcan be regarded as a crossed module in two obvious ways: ({1}, G, 1), where 1is the trivial map, or (G, G, idG). (ii) (G, {1},1) with the trivial action is a precrossed module. It satisfies the Peiffer identity if and only if Gis an abelian group. 18 1 Crossed modules and equivalent structures (iii) Given a G-module M, that is an abelian group Mtogether with an action of G on M,(M, G, 1) is a crossed module. (iv) If 1→N→H∂ −→ G→1is a central extension, that is a short exact sequence with N⊂Z(H), then (H, G, ∂)is a crossed module, with the action of Gon Hgiven by gh=hghh−1 gfor all g∈G,h∈H, where hgis an element in Hsuch that ∂(hg) = g, . The next one is a specially interesting example: Example 1.2.6. Let Hbe a group. The morphism α:H→Aut(H), where α(h)(h0) = hh0h−1, is a crossed module, together with the action of Aut(H)on Hdefined by ϕh=ϕ(h)for all ϕ∈Aut(H),h∈H. The interesting idea behind Aut(H) is not just (H, Aut(H), α) being a crossed module, but also the fact that for every action of a group Gon Hthere is a unique group homomorphism β:G→Aut(H) with gh=β(g)h. Therefore it would be possible to define a group action of Gon Has a group homomorphism from Gto Aut(H). The search for an analogous object in the categories of Lie algebras, Leibniz algebras and associative algebras led to the idea of actor in a category of interest [19, 20]. See Section 2.1 for more details. Definition 1.2.7. A morphism of crossed modules of groups (ϕ, ψ): (H, G, ∂)→ (H0, G0, ∂0)is a pair of group homomorphisms, ϕ:H→H0and ψ:G→G0, such that ψ∂ =∂0ϕ, (1.2.3) ϕ(gh) = ψ(g)ϕ(h),(1.2.4) for all g∈G,h∈H. Example 1.2.8. Let (H, G, ∂)be a crossed module of groups and Na group: (i) Given a morphism of groups ψ:G→Nsuch that ψ∂ = 0,(1, ψ): (H, G, ∂)→ ({1}, N, 1) is a morphism of crossed modules. (ii) Given a morphism of groups ψ:N→G,(1, ψ): ({1}, N, 1) →(H, G, ∂)is a morphism of crossed modules. (iii) Given a morphism of groups ψ:G→N,(ψ∂, ψ): (H, G, ∂)→(N, N, idN)is a morphism of crossed modules. In particular, (∂, idG): (H, G, ∂)→(G, G, idG)is a morphism of crossed modules. (iv) Given a morphism of groups ϕ:N→H,(ϕ, ∂ϕ): (N, N, idN)→(H, G, ∂)is a morphism of crossed modules. In particular, (idH, ∂): (H, H, idH)→(H, G, ∂)is a morphism of crossed modules. (v) Considering (G, Aut(G), α)as in Example 1.2.6, (∂, α): (H, G, ∂)→(G, Aut(G), α) is a morphism of crossed modules. 1.2.1 The case of groups 19 Composition of morphisms of crossed modules is defined component-wise and the identity morphism is given by (idH,idG) for any crossed module (H, G, ∂). We will denote by XGr the category of crossed modules of groups and morphisms of crossed modules. Bearing in mind Example 1.2.5 (i), it is possible to define the full embeddings E0:Gr →XGr and E1:Gr →XGr, where E0(G) = ({1}, G, 1) and E1(G) = (G, G, idG) for all G∈Gr. Given a morphism of groups α:G→G0,E0(α) = (1, α) and E1(α)=(α, α). Additionally, let us define the functors Υ0,Υ1and Υ2, from XGr to Gr, given by Υ0(H, G, ∂) = G/∂(H), Υ1(H, G, ∂) = Gand Υ2(H, G, ∂) = H, for any crossed module of groups (H, G, ∂). Given a morphism of crossed modules (ϕ, ψ): (H, G, ∂)→ (H0, G0, ∂0), Υ0(ϕ, ψ) = ψ,Υ1(ϕ, ψ) = ψand Υ2(ϕ, ψ) = ϕ, where ψis the morphism from G/∂(H) to G0/∂0(H0) induced by ψ. Note that ψis well defined, since given g1, g2∈Gsuch that g1=g2in G/∂(H), g1g−1 2∈∂(H), so there is h∈Hfor which ∂(h) = g1g−1 2. Due to (1.2.3), ψ(g1g−1 2) = ψ(∂(h)) = ∂0ϕ(h)∈∂0(H0). Proposition 1.2.9. Υ0is left adjoint to E0,E0is left adjoint to Υ1,Υ1is left adjoint to E1and E1is left adjoint to Υ2. Proof. It is fairly easy to construct the corresponding natural bijections in order to prove each of the adjunctions. Actually, there are explicit descriptions of them in Example 1.2.8 (i)–(iv). For instance, for the fist adjunction: Let (H, G, ∂) be a crossed module and Na group. Given α∈HomGr(G/∂(H), N), we can define the morphism of crossed modules H G {1}N ∂ 1απ 1 where πis the projection from Gto G/∂(H). Note that (1, απ) is a particular case of Example 1.2.8 (i). Conversely, given (1, ψ)∈HomXGr((H, G, ∂),({1}, N, 1)), due to (1.2.3), ψ∂ = 1. Hence, ˜ ψ:G/∂(H)→N, given by ˜ ψ(g) = ψ(g) is well defined. Naturality is obvious. The other three adjunctions can be proved similarly. Definition 1.2.10. Acat1-group (G1, G0, s, t)consists of a group G1together with a subgroup G0and structural the morphisms s, t:G1→G0such that s|G0=t|G0= idG0,(CGr1) [Ker s, Ker t] = 1,(CGr2) where [Ker s, Ker t]is the commutator of Ker sand Ker t. 20 1 Crossed modules and equivalent structures Definition 1.2.11. A morphism of cat1-groups γ: (G1, G0, s, t)→(G0 1, G0 0, s0, t0) is a group homomorphism γ:G1→G0 1such that γ(G0)⊆G0 0and s0γ=γ|G0s, t0γ=γ|G0t. Composition of morphisms of cat1-groups is obvious. We will denote by C1Gr the category of cat1-groups and morphisms of cat1-groups. Proposition 1.2.12. The categories XGr and C1Gr are equivalent. Proof. Given a crossed module of groups (H, G, ∂), the corresponding cat1-group is (HoG, G, s, t), where s(h, g) = gand t(h, g) = ∂(h)gfor all (h, g)∈HoG. It is clear that sis a group homomorphism, directly from the group operation in HoG and the definition of s. Regarding t, given (h1, g1),(h2, g2)∈HoG, t((h1, g1)(h2, g2)) = t(h1g1h2, g1g2) = ∂(h1g1h2)g1g2=∂(h1)g1∂(h2)g−1 1g1g2 =∂(h1)g1∂(h2)g2=t(h1, g1)t(h2, g2), due to (XGr1). Note that Gcan be regarded as a subgroup of HoGvia the monomorphism g7→ (1, g). It is clear that s|G=t|G= idG. Besides, Ker s={(h, 1) |h∈H} and Ker t={(h, ∂(h−1)) |h∈H}. Let h1, h2∈H. Directly from (XGr2), we have that h1∂(h−1 1)h2=h2h1. Therefore, (h2,1)(h1, ∂(h−1 1)) = (h2h1, ∂(h−1 1)) = (h1∂(h−1 1)h2, ∂(h−1 1)) = (h1, ∂(h−1 1))(h2,1), Hence [Ker s, Ker t] = 1 and (HoG, G, s, t) is a cat1-group. Additionally, given a morphism of crossed modules (ϕ, ψ): (H, G, ∂)→(H0, G0, ∂0), the corresponding morphism of cat1-groups is defined by fϕ,ψ(h, g)=(ϕ(h), ψ(g)) for any (h, g)∈HoG. It is clear that fϕ,ψ is a group homomorphism, directly from (1.2.4) and the fact that ϕand ψare group homomorphisms. On the other hand, it is immediate that fϕ,ψ(G)⊆G0, and the squares HoG G H0oG0G0 fϕ,ψ s fϕ,ψ|G s0 and HoG G H0oG0G0 fϕ,ψ t fϕ,ψ|G t0 are commutative, the first one directly from the definition of the morphisms involved; the second one due to (1.2.3). The previous assignments clearly define a functor from XGr to C1Gr, which will be denoted by catGr. Conversely, given a cat1-group (G1, G0, s, t), the corresponding crossed module is t|Ker s: Ker s→G0, with the action of G0on Ker sgiven by conjugation. Sometimes we will write simply tfor ease of notation. The action is obviously well defined. Regarding (XGr1), it follows immediately from (CGr1), specifically from the identity t|G0= idG0: t(yx) = t(yxy−1) = t(y)t(x)t(y−1) = yt(x)y−1 1.2.1 The case of groups 21 for all y∈G0,x∈Ker s. Now, let x1, x2∈Ker s. Since t|G0= idG0, it is clear that t(x−1 1)x1∈Ker t. Then, by (CGr2), x2t(x−1 1)x1=t(x−1 1)x1x2. Therefore t(x1)x2x1=t(x1)x2t(x−1 1)x1=t(x1)t(x−1 1)x1x2=x1x2, hence t(x1)x2=x1x2x−1 1, so (XGr2) holds. Moreover, given a morphism of cat1-groups γ: (G1, G0, s, t)→(G0 1, G0 0, s0, t0), its corresponding morphism of crossed modules is given by Ker s G0 Ker s0G0 0. γ|Ker s t|Ker s γ|G0 t0|Ker s0 Note that γ(Ker s)⊂Ker s0, directly from the identity s0γ=γ|G0s. The commutativity of the previous diagram follows from the identity t0γ=γ|G0t. Besides, given y∈G0,x∈Ker s, the identity γ|Ker s(yx) = γ|G0(y)γ(x) follows from the definition of the action of G0on Ker sand the fact that γis a group homomorphism. The previous assignments clearly define a functor from C1Gr to XGr, which will be denoted by XmGr. catGr and XmGr establish an equivalence between the categories XGr and C1Gr, with the natural isomorphisms α:1XGr →XmGr ◦catGr and β:1C1Gr →catGr ◦XmGr given, for a fixed (H, G, ∂) in XGr and a fixed (G1, G0, s, t) in C1Gr, by: H G Ho{1}G αH ∂ idG ∂ and G1G0 Ker soG0G0 βG1 t s idG0 ˜ t ˜s respectively, where αH(h) = (h, 1) for every h∈H,βG1(g) = (gs(g−1), s(g)) for every g∈G1. It is clear that (αH,idG) is an isomorphism of crossed modules and the naturality of αis obvious. Concerning βG1, observe that catGr(XmGr(G1, G0, s, t)) = (Ker soG0, G0,˜s, ˜ t), with ˜s(x, y) = yand ˜ t(x, y) = t(x)yfor all x∈Ker s,y∈G0. It is easy to check that βG1is a group homomorphism just by using the definition of the group operation in Ker soG0and the action of G0on Ker s. Besides, given y∈G0,βG1(y) = (1, y), since s|G0= idG0. Calculations in order to check the identities ˜sβG1=sand ˜ tβG1=t are obvious. The inverse of βG1is given by β−1 G1(x, y) = xy, for all (x, y)∈Ker soG0. Naturality of βcan be readily checked by using the identity s0γ=γ|G0sfor any morphism γbetween two given cat1-groups (G1, G0, s, t) and (G0 1, G0 0, s0, t0). 22 1 Crossed modules and equivalent structures 1.2.2 The case of Lie algebras Lie crossed modules have been investigated by various authors. Namely, in [60] Kassel and Loday use Lie crossed modules as computational tools in order to give an interpretation of the third relative Chevalley-Eilenberg cohomology of Lie algebras. Guin [51] developed the low-dimensional non-abelian cohomology of Lie algebras with coefficients in Lie crossed modules, which later was extended to higher dimensions in [57]. Internal (cotriple) homology and Chevalley-Eilenberg homology theories of Lie crossed modules were investigated in [23, 35]. Lie crossed modules also occur in the “categorification” problem of the theory of Lie algebras [4] as an equivalent formulation of strict Lie 2-algebras (see Section 1.3). Recall that a Lie algebra pover Kis a K-module together with a bilinear operation [,]: p×p→p, called the Lie bracket, such that [p, p]=0, [p1,[p2, p3]] + [p2,[p3, p1]] + [p3,[p1, p2]] = 0, for all p, p1, p2, p3∈p. The second equality is usually called the Jacobi identity. Amorphism of Lie algebras is a K-linear map that preserves the bracket. We will denote by Lie the category of Lie algebras and morphisms of Lie algebras. Definition 1.2.13. Let mand pbe two Lie algebras. An action of pon mis a bilinear map p×m→m,(p, m)7→ [p, m]such that (1) [[p1, p2], m]=[p1,[p2, m]] −[p2,[p1, m]], (2) [p, [m1, m2]] = [[p, m1], m2]+[m1,[p, m2]], for all m, m1, m2∈mand p, p1, p2∈p. We will say that pacts trivially on mif [p, m] = 0 for all m∈m,p∈p. Observe that we denote the action by the same symbol used for the multiplication in mand p, by analogy to the notation used for Ω-groups. This means no ambiguity, since the arguments of the bracket will always determine the only possible choice. Note that the two identities in the definition of a Lie action can be obtained from the Jacobi identity by taking two elements in pand one in m(first identity), and two elements in mand one in p(second identity). Given a Lie action of pon mwe can form the semidirect product Lie algebra, mop, with the underlying K-module m⊕pand the Lie bracket given by [(m1, p1),(m2, p2)] = ([m1, m2]+[p1, m2]−[p2, m1],[p1, p2]), for all (m1, p1),(m2, p2)∈m⊕p. 1.2.2 The case of Lie algebras 23 Definition 1.2.14. A crossed module of Lie algebras (or Lie crossed module) (m,p, ν) is a Lie homomorphism ν:m→ptogether with an action of pon msuch that ν([p, m]) = [p, ν(m)],(XLie1) [ν(m1), m2]=[m1, m2].(XLie2) for all m, m1, m2∈mand p∈p. For the sake of coherence, (XLie1) will be called equivariance and (XLie2) Peiffer identity. If (m,p, ν) satisfies (XLie1) but not necessarily (XLie2), it is called precrossed module. Moreover, we have the following result: Lemma 1.2.15. Given a Lie crossed module (m,p, ν), (i) Ker νis an ideal of mand Im νis an ideal of p. (ii) Ker ν⊂Ann(m), where Ann(m)is the annihilator of m. Proof. Ker νis an ideal for any Lie homomorphism ν. The calculations for the other two statements can be easily completed by using (XLie1) and (XLie2) respectively. Example 1.2.16. Let pbe a Lie algebra. (i) The Lie bracket in pyields an action of pon any ideal qof p. The inclusion i:q,→ptogether with that action is a Lie crossed module. {0}and pare ideals of p, so any Lie algebra pcan be regarded as a crossed module in two obvious ways: ({0},p,0), where 0is the trivial map, or (p,p,idp). (ii) (p,{0},0) with the trivial action is a precrossed module. It satisfies the Peiffer identity if and only if pis abelian, that is, if the bracket is trivial. (iii) If 0→q→mν −→ p→0is a short exact sequence with q⊂Ann(m), then (m,p, ν) is a Lie crossed module, with the action of pon mgiven by [p, m] = [mp, m]for all p∈p,m∈m, where mpis an element in msuch that ν(mp) = p. The role played by Aut(H) for any group H(see Example 1.2.6) is played by Der(m), the Lie algebra of derivations, for any Lie algebra m. Note that an element d∈ Der(m) is a K-linear map from mto msuch that d[m1, m2]=[d(m1), m2]+[m1, d(m2)]. The Lie structure in Der(m) is given by [d1, d2] = d1d2−d2d1for all d1, d2∈Der(m). Example 1.2.17. The Lie homomorphism α:m→Der(m), where α(m)(m0) = [m, m0], together with the action of Der(m)on mdefined by [ϕ, m] = ϕ(m)for all ϕ∈Der(m),m∈m, is a Lie crossed module. Every action of a Lie algebra pon a Lie algebra myields a unique morphism of Lie algebras β:p→Der(m), such that [β(p), m]=[p, m]. Hence, it would be possible to define a Lie action of pon mas a Lie homomorphism from pto Der(m). See Section 2.1 for more details. 24 1 Crossed modules and equivalent structures Definition 1.2.18. A morphism of Lie crossed modules (ϕ, ψ): (m,p, ν)→(m0,p0, ν0) is a pair of Lie homomorphisms, ϕ:m→m0and ψ:p→p0, such that ψν =ν0ϕ, (1.2.5) ϕ([p, m]) = [ψ(p), ϕ(m)],(1.2.6) for all m∈m,p∈p. Example 1.2.19. Let (m,p, ν)be a Lie crossed module and qa Lie algebra. (i) Given a Lie homomorphism ψ:p→qsuch that ψν = 0,(0, ψ): (m,p, ν)→ ({0},q,0) is a morphism of Lie crossed modules. (ii) Given a Lie homomorphism ψ:q→p,(0, ψ): ({0},q,0) →(m,p, ν)is a morphism of Lie crossed modules. (iii) Given a Lie homomorphism ψ:p→q,(ψν, ψ): (m,p, ν)→(q,q,idq)is a morphism of Lie crossed modules. In particular, (ν, idp): (m,p, ν)→(p,p,idp)is a morphism of Lie crossed modules. (iv) Given a Lie homomorphism ϕ:q→m,(ϕ, νϕ): (q,q,idq)→(m,p, ν)is a morphism of Lie crossed modules. In particular, (idm, ν): (m,m,idm)→(m,p, ν)is a morphism of Lie crossed modules. (v) (ν, α): (m,p, ν)→(p,Der(p), α)is a morphism of Lie crossed modules, with (p,Der(p), α)as in Example 1.2.17. Composition of morphisms of Lie crossed modules is defined component-wise and the identity morphism is given by (idm,idp) for any crossed module (m,p, ν). We will denote by XLie the category of Lie crossed modules and morphisms of Lie crossed modules. Just like in the case of crossed modules of groups, it is possible to define the full embeddings I0:Lie →XLie and I1:Lie →XLie, with I0(p)=({0},p,0) and I1(p) = (p,p,idp) for any Lie algebra p. Given a morphism of Lie algebras α:p→p0, I0(α) = (0, α) and I1(α) = (α, α). Continuing with the analogy, let us define the functors Φ0,Φ1and Φ2, from XLie to Lie, given by Φ0(m,p, ν) = p/ν(m), Φ1(m,p, ν) = pand Φ2(m,p, ν) = mfor any Lie crossed module (m,p, ν). Given a morphism of Lie crossed modules (ϕ, ψ): (m,p, ν)→ (m0,p0, ν0), Φ0(ϕ, ψ) = ψ,Φ1(ϕ, ψ) = ψand Φ2(ϕ, ψ) = ϕ, where ψis the morphism from p/ν(m) to p0/ν0(m0) induced by ψ. Proposition 1.2.20. Φ0is left adjoint to I0,I0is left adjoint to Φ1,Φ1is left adjoint to I1and I1is left adjoint to Φ2. Proof. The corresponding natural bijections can be readily described by using Example 1.2.19 (i)–(iv). Definition 1.2.21. Acat1-Lie algebra (p1,p0, s, t)consists of a Lie algebra p1together with a Lie subalgebra p0and the structural morphisms s, t:p1→p0such 1.2.2 The case of Lie algebras 25 that s|p0=t|p0= idp0,(CLie1) [Ker s, Ker t]=0.(CLie2) Definition 1.2.22. A morphism of cat1-Lie algebras γ: (p1,p0, s, t)→(p0 1,p0 0, s0, t0) is a Lie homomorphism γ:p1→p0 1such that γ(p0)⊆p0 0and s0γ=γ|p0s,t0γ= γ|p0t. Composition of morphisms of cat1-Lie algebras is obvious. We will denote by C1Lie the category of cat1-Lie algebras and morphisms of cat1-Lie algebras. Proposition 1.2.23. The categories XLie and C1Lie are equivalent. Proof. Given a crossed module of Lie algebras (m,p, ν), the corresponding cat1-Lie algebra is (mop,p, s, t), with s(m, p) = pand t(m, p) = ν(m)+pfor all (m, p)∈mop. It is evident that sis a Lie homomorphism, while tpreserves the bracket due to (XLie1), the fact that νis a Lie homomorphism and the bilinearity and antisymmetry of the bracket in p. Note that pcan be regarded as a Lie subalgebra of mopvia the morphism p7→ (0, p). It is obvious that s|p=t|p= idp. Directly from the definition of sand t, we get that Ker s={(m, 0) |m∈m}and Ker t={(m, −ν(m)) |m∈m}. Given m1, m2∈m, due to (XLie2) and the antisymmetry of the bracket in m, we know that −[m1, m2] = [m2, m1]=[ν(m2), m1]. Hence, [(m1,0),(m2,−ν(m2))] = ([m1, m2]+[ν(m2), m1],0) = (0,0). Therefore, [Ker s, Ker t] = 0 and (mop,p, s, t) is a cat1-Lie algebra. Additionally, given a morphism of Lie crossed modules (ϕ, ψ) from (m,p, ν) to (m0,p0, ν0), the corresponding morphism of cat1-Lie algebras is defined by fϕ,ψ(m, p) = (ϕ(m), ψ(p)) for all (m, p)∈mop. One can easily check that fϕ,ψ is a Lie homomorphism by making use of (1.2.6) and the fact that ϕand ψare Lie homomorphisms. It is clear that fϕ,ψ(p)⊆p0. Besides, the identity s0γ=γ|p0sfollows from the definition of the morphisms involved, while t0γ=γ|p0tis an immediate consequence of (1.2.5). The previous assignments clearly define a functor from XLie to C1Lie, which will be denoted by catLie. Conversely, given a cat1-Lie algebra (p1,p0, s, t), the corresponding Lie crossed module is t|Ker s: Ker s→p0, with the action of p0on Ker sinduced by the bracket in p1. We will write simply tinstead of t|Ker s. (XLie1) follows directly from the fact that tis a Lie homomorphism and (CLie1), specifically from the identity t|p= idp. Now, let x1, x2∈Ker s. It is clear that t(x1)−x1∈Ker t, since tis linear and t|p0= idp0. Therefore, due to (CLie2) and the bilinearity of the bracket in p1, we have that 0=[t(x1)−x1, x2] = [t(x1), x2]−[x1, x2]. Hence, (Ker s, p0, t|Ker s) satisfies (XLie2) and it is a Lie crossed module. 32 1 Crossed modules and equivalent structures If the bracket of a Leibniz algebra phappens to be anticommutative, the pis a Lie algebra. Furthermore, every Lie algebra is a Leibniz algebra. We will denote by Lb the category of Leibniz algebras and morphisms of Leibniz algebras. Definition 1.2.37. Let pand mbe two Leibniz algebras. An action of pon mconsists of a pair of bilinear maps, p×m→m,(p, m)7→ [p, m]and m×p→m,(m, p)7→ [m, p], such that (1) [p, [m1, m2]] = [[p, m1], m2]−[[p, m2], m1], (2) [m1,[p, m2]] = [[m1, p], m2]−[[m1, m2], p], (3) [m1,[m2, p]] = [[m1, m2], p]−[[m1, p], m2], (4) [m, [p1, p2]] = [[m, p1], p2]−[[m, p2], p1], (5) [p1,[m, p2]] = [[p1, m], p2]−[[p1, p2], m], (6) [p1,[p2, m]] = [[p1, p2], m]−[[p1, m], p2], for all m, m1, m2∈mand p, p1, p2∈p. Note that, as an immediate consequence of (2) and (3), [m1,[p, m2]] + [m1,[m2, p]] = 0 for all m1, m2∈mand p∈p. On the other hand, from (5) and (6), [p1,[m, p2]] + [p1,[p2, m]] = 0 for all m∈mand p1, p2∈p. We will say that pacts trivially on mif [p, m] = 0 = [m, p] for all m∈m,p∈p. Observe that we denote the action by the same symbol used for the multiplication in mand p, by analogy to the notation used for Ω-groups. Note that the six identities in the definition of a Leibniz action can be obtained from the Leibniz identity by taking two elements in pand one in m(three identities), and two elements in mand one in p(the other three identities). For example, if pis a Leibniz subalgebra of some Leibniz algebra q, and if mis an ideal in q, then the Leibniz bracket in qyields an action of pon m. Given a Leibniz action of pon mwe can consider the semidirect product Leibniz algebra mop, which consists of the K-module m⊕ptogether with the Leibniz bracket given by [(m1, p1),(m2, p2)] = ([m1, m2]+[p1, m2]+[m1, p2],[p1, p2]) for all (m1, p1),(m2, p2)∈m⊕p. 1.2.4 The case of Leibniz algebras 33 Definition 1.2.38. A crossed module of Leibniz algebras (or Leibniz crossed module) (m,p, η)is a morphism of Leibniz algebras η:m→ptogether with an action of pon msuch that η([p, m]) = [p, η(m)] and η([m, p]) = [η(m), p],(XLb1) [η(m1), m2]=[m1, m2] = [m1, η(m2)].(XLb2) for all m, m1, m2∈m,p∈p. For the sake of coherence, (XLb1) will be called equivariance and (XLb2) Peiffer identity. If (m,p, η) satisfies (XLb1) but not necessarily (XLb2), it is called precrossed module. Moreover, we have the following result: Lemma 1.2.39. Given a Leibniz crossed module (m,p, η), (i) Ker ηis an ideal of mand Im ηis an ideal of p. (ii) Ker η⊂Ann(m), where Ann(m)is the annihilator of m. Example 1.2.40. Let pbe a Leibniz algebra. (i) The Leibniz bracket in pyields an action of pon any ideal qof p. The inclusion i:q,→ptogether with that action is a Leibniz crossed module. {0}and pare ideals of p, so any Leibniz algebra pcan be regarded as a crossed module in two obvious ways: ({0},p,0), where 0is the trivial map, or (p,p,idp). (ii) (p,{0},0) with the trivial action is a precrossed module. It satisfies the Peiffer identity if and only if the bracket in pis trivial. (iii) If 0→q→mη −→ p→0is a short exact sequence with q⊂Ann(m), then (m,p, η) is a Leibniz crossed module, with the action of pon mgiven by [p, m] = [mp, m]for all m∈m,p∈p, where mpis an element in msuch that η(mp) = p. The analogue to (H, Aut(H), α) in Gr and (m,Der(m), α) in Lie (see Examples 1.2.6 and 1.2.17) does not always exist in Lb. We will give more details about this construction in Section 2.1. Definition 1.2.41. A morphism of Leibniz crossed modules (ϕ, ψ)from (m,p, η)to (m0,p0, η0)is a pair of Leibniz homomorphisms, ϕ:m→m0and ψ:p→p0, such that ψη =η0ϕ, (1.2.9) ϕ([p, m]) = [ψ(p), ϕ(m)] and ϕ([m, p]) = [ϕ(m), ψ(p)] (1.2.10) for all m∈m,p∈p. Example 1.2.42. Let (m,p, η)be a Leibniz crossed module and qa Leibniz algebra. (i) Given a Leibniz homomorphism ψ:p→qsuch that ψη = 0,(0, ψ): (m,p, η)→ ({0},q,0) is a morphism of Leibniz crossed modules. 34 1 Crossed modules and equivalent structures (ii) Given a Leibniz homomorphism ψ:q→p,(0, ψ): ({0},q,0) →(m,p, η)is a morphism of Leibniz crossed modules. (iii) Given a Leibniz homomorphism ψ:p→q,(ψη, ψ): (m,p, η)→(q,q,idq)is a morphism of Leibniz crossed modules. In particular, (η, idp): (m,p, η)→(p,p,idp)is a morphism of Leibniz crossed modules. (iv) Given a Leibniz homomorphism ϕ:q→m,(ϕ, ηϕ): (q,q,idq)→(m,p, η)is a morphism of Leibniz crossed modules. In particular, (idm, η): (m,m,idm)→(m,p, η) is a morphism of Leibniz crossed modules. Composition of morphisms of Leibniz crossed modules is defined component-wise and the identity morphism is given by (idm,idp) for any Leibniz crossed module (m,p, η). We will denote by XLb the category of Leibniz crossed modules and morphisms of Leibniz crossed modules. Just like in the case of crossed modules of groups, it is possible to define the full embeddings J0:Lb →XLb and J1:Lb →XLb, with J0(p) = ({0},p,0) and J1(p) = (p,p,idp) for any Leibniz algebra p. Given a morphism of Leibniz algebras α:p→p0,J0(α) = (0, α) and J1(α)=(α, α). Besides, let us define the functors Ψ0,Ψ1and Ψ2, from XLb to Lb, given by Ψ0(m,p, η) = p/η(m), Ψ1(m,p, η) = pand Ψ2(m,p, η) = mfor any Leibniz crossed module (m,p, η). Given a morphism of Leibniz crossed modules (ϕ, ψ): (m,p, η)→ (m0,p0, η0), Ψ0(ϕ, ψ) = ψ,Ψ1(ϕ, ψ) = ψand Ψ2(ϕ, ψ) = ϕ, where ψis the morphism from p/η(m) to p0/η0(m0) induced by ψ. Proposition 1.2.43. Ψ0is left adjoint to J0,J0is left adjoint to Ψ1,Ψ1is left adjoint to J1and J1is left adjoint to Ψ2. Proof. The corresponding natural bijections can be readily described by using Example 1.2.42 (i)–(iv). Definition 1.2.44. Acat1-Leibniz algebra (p1,p0, s, t)consists of a Leibniz algebra p1together with a Leibniz subalgebra p0and the structural morphisms s, t:p1→p0 such that s|p0=t|p0= idp0,(CLb1) [Ker s, Ker t] = 0 = [Ker t, Ker s] (CLb2) Definition 1.2.45. A homomorphism of cat1-Leibniz algebras γfrom (p1,p0, s, t) to (p0 1,p0 0, s0, t0)is a Leibniz homomorphism γ:p1→p0 1such that γ(p0)⊆p0 0and s0γ=γ|p0s,t0γ=γ|p0t. Composition of morphisms of cat1-Lie algebras is obvious. We will denote by C1Lb the category of cat1-Leibniz algebras and morphisms of cat1-Leibniz algebras. Proposition 1.2.46. The categories XLb and C1Lb are equivalent. 1.2.4 The case of Leibniz algebras 35 Proof. Given a crossed module of Leibniz algebras (m,p, η), the corresponding cat1Leibniz algebra is (mop,p, s, t), where s(m, p) = pand t(m, p) = η(m) + pfor all (m, p)∈mop. It is evident that sis a Leibniz homomorphism, while tis a Leibniz homomorphism due to (XLb1), the fact that ηis a Leibniz homomorphism and the bilinearity of the bracket in p. Note that pcan be regarded as a Lie subalgebra of mopvia the morphism p7→ (0, p). It is clear that s|p=t|p= idp. Directly from the definition of sand t, we get that Ker s={(m, 0) |m∈m}and Ker t={(m, −η(m)) | m∈m}. Let (m1,0) ∈Ker sand (m2,−η(m2)) ∈Ker t: [(m1,0),(m2,−η(m2))] = ([m1, m2]−[m1, η(m2)] ,0) = (0,0), due to (XLb2). Analogously [(m2,−η(m2)),(m1,0)] = (0,0), so [Ker s, Ker t]=0= [Ker t, Ker s] and (mop,p, s, t) is a cat1-Leibniz algebra. Additionally, given a morphism of Leibniz crossed modules (ϕ, ψ) from (m,p, η) to (m0,p0, η0), the corresponding morphism of cat1-Leibniz algebras is defined by fϕ,ψ(m, p) = (ϕ(m), ψ(p)), for all (m, p)∈mop. One can easily check that fϕ,ψ is a Leibniz homomorphism by making use of (1.2.10) and the fact that ϕand ψ are Leibniz homomorphisms. It is clear that fϕ,ψ(p)⊆p0. The identity s0γ=γ|p0s follows from the definition of the morphisms involved and t0γ=γ|p0tis an immediate consequence of (1.2.9). The previous assignments clearly define a functor from XLb to C1Lb, which will be denoted by catLb. Conversely, given a cat1-Leibniz algebra (p1,p0, s, t), the corresponding Leibniz crossed module is t|Ker s: Ker s→p0, with the action of p0on Ker sinduced by the bracket in p1. (XLb1) follows from the fact that tis a Leibniz homomorphism and (CLb1), specifically from the identity t|p0= id p. Now, let x1, x2∈Ker s. It is clear that t(x1)−x1∈Ker t, since tis linear and t|p0= idp0. Therefore, due to (CLb2) and the bilinearity of the bracket in p1, we have that 0=[t(x1)−x1, x2] = [t(x1), x2]−[x1, x2]. Analogously, [x1, t(x2)] = [x1, x2]. Thus, (Ker s, p0, t|Ker s) is a crossed module of Leibniz algebras. Given a morphism of cat1-Leibniz algebras γ: (p1,p0, s, t)→(p0 1,p0 0, s0, t0), its corresponding morphism of Leibniz crossed modules is given by Ker sp0 Ker s0p0 0. γ|Ker s t|Ker s γ|p0 t0|Ker s0 Note that γ(Ker s)⊂Ker s0, directly from the identity s0γ=γ|p0s. The commutativity of the previous diagram follows from the identity t0γ=γ|p0t. Besides, (1.2.10) follows from the definition of the action of p0on Ker sand the fact that γis a Leibniz 36 1 Crossed modules and equivalent structures homomorphism. The previous assignments clearly define a functor from C1Lb to XLb, which will be denoted by XmLb. catLb and XmLb establish an equivalence between the categories XLb and C1Lb, with the natural isomorphisms α:1XLb →XmLb ◦catLb and β:1C1Lb →catLb ◦XmLb given, for a fixed (m,p, η) in XLb and a fixed (p1,p0, s, t) in C1Lb, by: m p mo{0}p αm η idp η and p1p0 Ker sop0p0 βp1 t s idp0 ˜ t ˜s respectively, where αm(m) = (m, 0) for every m∈m,βp1(p) = (p−s(p), s(p)) for every p∈p1. It is clear that (αm,idp) is an isomorphism of Leibniz crossed modules and the naturality of αis obvious. Concerning βp1, observe that catLb(XmLb(p1,p0, s, t)) = (Ker sop0,p0,˜s, ˜ t), with ˜s(x, y) = yand ˜ t(x, y) = t(x)+yfor all x∈Ker s,y∈p0. It is easy to check that βp1is a Leibniz homomorphism just by using the definition of the Leibniz bracket in Ker so p0and the action of p0on Ker s. Besides, given y∈p0,βp1(y) = (0, y), since s|p0= idp0. Calculations in order to check the identities ˜sβp1=sand ˜ tβp1=tare obvious. The inverse of βp1is given by β−1 p1(x, y) = x+y, for all (x, y)∈Ker sop0. Naturality of βcan be readily checked by using the identity s0γ=γ|p0sfor any morphism γ between two given cat1-Leibniz algebras (p1,p0, s, t) and (p0 1,p0 0, s0, t0). 1.2.5 The case of dialgebras Associative dialgebras (or simply dialgebras), also known as diassociative algebras, were introduced by Loday [65] as an algebraic structure with a role with respect to Leibniz algebras analogous to the one that associative algebras play with respect to Lie algebras. Let us recall some basic definitions and elemental properties from [65]. Definition 1.2.47 ([65]).An associative dialgebra (or simply dialgebra), is a Kmodule Dequipped with two K-linear maps a,`:D⊗D→D, called the left product and the right product respectively, satisfying the following axioms (xay)az=xa(y`z),(Di1) (xay)az=xa(yaz),(Di2) (x`y)az=x`(yaz),(Di3) (xay)`z=x`(y`z),(Di4) (x`y)`z=x`(y`z),(Di5) 1.2.5 The case of dialgebras 37 for all x, y, z ∈D. A morphism of dialgebras is a K-linear map that preserves both the left and the right products. Observe that here we break the “rule” of denoting elements with lower case letters after the name of the object (Din this particular case). This decision was made in order to prevent confusions with derivations. Likewise, whenever we consider a dialgebra L, its elements will not be denoted by l, since that symbol will be used for a specific type of maps in Subsection 2.1.1. We will denote by Dias the category of dialgebras and morphisms of dialgebras. Remark 1.2.48. In some identities we may use ∗to denote both `and a, meaning that the corresponding equality is satisfied for ∗=`and ∗=a. Abar-unit in a dialgebra Dis an element e∈Dsuch that xae=x=e`x for all x∈D. A bar-unit is not necessarily unique. The set of all bar-units is called halo. A unital dialgebra is a dialgebra with a specific bar-unit e. A morphism of dialgebras is said to be unital if the image of any bar-unit is a bar-unit. Observe that if a dialgebra has a unit , i.e. an element such that ax=xfor all x∈D, from (Di1) we have (ay)az=a(y`z), that is yaz=y`zfor y, z ∈D. Hence a=`and Dis merely an associative algebra with unit. A dialgebra Dis called abelian if both the left and the right products are trivial, that is xay=x`y= 0 for all x, y ∈D. Note that any K-module can be regarded as an abelian dialgebra. A submodule Iof a dialgebra Dis called an ideal of Dif xay, x `y, y ax, y `x∈Ifor any x∈Iand y∈D. The annihilator of a dialgebra Dis given by: Ann(D) = {x∈D|xay=yax=x`y=y`x= 0,for all y∈D}. It is immediate to check that Ann(D) is indeed an ideal of D. Example 1.2.49 ([65]). (i) Any algebra Awith aab=a·b=a`bfor all a, b ∈Ais a dialgebra. (ii) Let (A, d)be a non-graded differential associative algebra. By hypothesis, d(a·b) = d(a)·b+a·d(b)and d2= 0.Ais a dialgebra with the left and the right products given, for any a, b ∈A, by aab=a·d(b)and a`b=d(a)·b (iii) Let Abe an associative algebra and Mbe an A-bimodule. Let f:M→Abe an A-bimodule map. Mtogether with man=m·f(n)and m`n=f(m)·n 38 1 Crossed modules and equivalent structures for all m, n ∈M, is a dialgebra. (iv) Let Abe an associative algebra and put D=A⊕A. The products a1⊕b1aa2⊕b2=a1⊕b1·a2·b2and a1⊕b1`a2⊕b2=a1·b1·a2⊕b2, extended by linearity to A⊕A, endow it with a dialgebra structure. Another interesting dialgebra is studied by Lin and Zhang in [62]. Let F[x, y] be the polynomial algebra over a field Fof characteristic zero. Then F[x, y] is a dialgebra with the products f(x, y)ag(x, y) = f(x, y)·g(y, y) and f(x, y)`g(x, y) = f(x, x)·g(x, y), for all f(x, y), g(x, y)∈F[x, y]. It is not mentioned in [62], but this structure can be extended to the polynomial algebra with nvariables F[x1, . . . , xn] by defining the products as f(x1, . . . , xn)ag(x1, . . . , xn) = f(x1, . . . , xn)·g(xn, . . . , xn), f(x1, . . . , xn)`g(x1, . . . , xn) = f(x1, . . . , x1)·g(x1, . . . , xn). Definition 1.2.50. Let Dand Lbe dialgebras. An action of Don Lconsists of four linear maps, two of them denoted by the symbol aand the other two by `, a:D⊗L→L, a:L⊗D→L, `:D⊗L→L, `:L⊗D→L such that the following 30 equalities hold: (1) (xaa)ab=xa(a`b), (2) (xaa)ab=xa(aab), (3) (x`a)ab=x`(aab), (4) (xaa)`b=x`(a`b), (5) (x`a)`b=x`(a`b), (6) (aax)ab=aa(x`b), (7) (aax)ab=aa(xab), (8) (a`x)ab=a`(xab), (9) (aax)`b=a`(x`b), (10) (a`x)`b=a`(x`b), (11) (aab)ax=aa(b`x), (12) (aab)ax=aa(bax), (13) (a`b)ax=a`(bax), (14) (aab)`x=a`(b`x), (15) (a`b)`x=a`(b`x), (16) (aax)ay=aa(x`y), (17) (aax)ay=aa(xay), (18) (a`x)ay=a`(xay), (19) (aax)`y=a`(x`y), (20) (a`x)`y=a`(x`y), 1.2.5 The case of dialgebras 39 (21) (xaa)ay=xa(a`y), (22) (xaa)ay=xa(aay), (23) (x`a)ay=x`(aay), (24) (xaa)`y=x`(a`y), (25) (x`a)`y=x`(a`y), (26) (xay)aa=xa(y`a), (27) (xay)aa=xa(yaa), (28) (x`y)aa=x`(yaa), (29) (xay)`a=x`(y`a), (30) (x`y)`a=x`(y`a), for all x, y ∈D;a, b ∈L. The action is called trivial if these four maps are trivial. Note that the previous identities are obtained from the axioms (Di1)–(Di5) by taking one variable in Dand two variables in L(15 equalities), and one variable in Land two variables in D(15 equalities). Observe that we denote the action by the same symbol used for the left and the right products in Dand L, by analogy to the notation used for Ω-groups. Let us show some examples of actions. Note that Dias is a category of interest and the first example agrees with the definition of a set of derived actions from a D-structure. Examples 1.2.51. (i)If 0→Lι →Eσ →D→0is a split short exact sequence of dialgebras, i.e. there exists a homomorphism of dialgebras ϕ:D→Esuch that σϕ = idD, then there is an action of the dialgebra Don Ldefined in the standard way by taking the left and the right products in the dialgebra E: x∗a=ϕ(x)∗ι(a)and a∗x=ι(a)∗ϕ(x) for any x∈D,a∈L. (ii)If Dis a subdialgebra of a dialgebra E(maybe D=E) and Iis an ideal in E, then the left and the right products in Eyield an action of Don I. (iii)Any morphism of dialgebras D→Linduces an action of Don Lin the standard way by taking images of elements of Dand the left and the right products in L. (iv)If µ:L→Dis an surjective morphism of dialgebras with Ker µin the annihilator of L, then there is an action of Don Ldefined in the standard way by taking preimages of the elements of Dand the left and the right products in L. (v)If Lis a bimodule over a dialgebra D(for the definition see [65, Subsection 2.3]), thought as an abelian dialgebra, then the bimodule structure defines an action of D on the (abelian) dialgebra L. Note that if a dialgebra Dacts on a dialgebra L, then L, as a K-module, has a structure of bimodule over the dialgebra D. Given an action of a dialgebra Don a dialgebra Lwe define the semidirect product dialgebra, LoD, with the underlying K-module L⊕Dendowed with the left and 40 1 Crossed modules and equivalent structures the right products given by (a1, x1)∗(a2, x2) = (a1∗a2+x1∗a2+a1∗x2, x1∗x2). for all x1, x2∈D,a1, a2∈L. Definition 1.2.52. A crossed module of dialgebras (L, D, µ)is a morphism of dialgebras µ:L→Dtogether with an action of Don Lsuch that µ(x∗a) = x∗µ(a)and µ(a∗x) = µ(a)∗x, (XDi1) µ(a1)∗a2=a1∗a2=a1∗µ(a2).(XDi2) for all x∈D,a1, a2∈L. For the sake of coherence, (XDi1) will be called equivariance and (XDi2) Peiffer identity. If (L, D, µ) satisfies (XDi1) but not necessarily (XDi2), it is called precrossed module. Moreover, we have the following result: Lemma 1.2.53. Given a crossed module of dialgebras (L, D, µ), (i) Ker µis an ideal of Land Im µis an ideal of D. (ii) Ker µ⊂Ann(L). The first two examples immediately below show that the concept of crossed module of dialgebras generalizes both the concepts of ideal and bimodule of dialgebras. Example 1.2.54. Let Dbe a dialgebra. (i)The inclusion L ,→Dof an ideal Lof Dis a crossed module, with the action of Don Lis given by the left and the right products in D, as in Example 1.2.51 (ii). Conversely, if µ:L→Dis a crossed module of dialgebras which is injective, then by Lemma 1.2.53 (i),Lis isomorphic to an ideal of D.{0}and Dare ideals of D, so any dialgebra Dcan be regarded as a crossed module in two obvious ways: ({0}, D, 0), where 0is the trivial map, or (D, D, idD). (ii)For any bimodule Lover a dialgebra Dthe trivial map 0 : L→Dis a crossed module with the action of Don the (abelian) dialgebra Ldescribed in Example 1.2.51 (v). Conversely, if 0 : L→Dis a crossed module of dialgebras, then Lis necessarily an abelian dialgebra and the action of Don Ldetermines on La bimodule structure over D. (iii)Any morphism of dialgebras µ:L→Dwith Labelian and Im µ⊂Ann(D)is a crossed module together with the trivial action of Don L. (iv)Any surjective morphism of dialgebras µ:L→Dwith Ker µ⊂Ann(L)and the action of Don Ldescribed in Example 1.2.51 (iv) is a crossed module of dialgebras. The analogue to (H, Aut(H), α) in Gr and (m,Der(m), α) in Lie (see Examples 1.2.6 and 1.2.17) does not always exist in Dias. We will give more details about this construction in Section 2.1. 1.2.5 The case of dialgebras 41 Definition 1.2.55. Amorphism of crossed modules of dialgebras (ϕ, ψ): (L, D, µ)→ (L0, D0, µ0) is a pair of dialgebra homomorphisms, ϕ:L→L0and ψ:D→D0, such that ψµ =µ0ϕ, (1.2.11) ϕ(x∗a) = ψ(x)∗ϕ(a) and ϕ(a∗x) = ϕ(a)∗ψ(x),(1.2.12) for all x∈D,a∈L. Example 1.2.56. Let (L, D, µ)be a crossed module of algebras and Ea dialgebra. (i) Given a morphism of dialgebras ψ:D→Esuch that ψµ = 0,(0, ψ): (L, D, µ)→ ({0}, E, 0) is a morphism of crossed modules. (ii) Given a morphism of dialgebras ψ:E→D,(0, ψ): ({0}, E, 0) →(L, D, µ)is a morphism of crossed modules. (iii) Given a morphism of dialgebras ψ:D→E,(ψµ, ψ): (L, D, µ)→(E, E, idE)is a morphism of crossed modules. In particular, (µ, idD): (L, D, µ)→(D, D, idD)is a morphism of crossed modules. (iv) Given a morphism of dialgebras ϕ:E→L,(ϕ, µϕ): (E, E, idE)→(L, D, µ)is a morphism of crossed modules. In particular, (idL, µ): (L, L, idL)→(L, D, µ)is a morphism of crossed modules. Composition of morphisms of crossed modules of dialgebras is defined componentwise and the identity morphism is given by (idL,idD) for any crossed module (L, D, µ). We will denote by XDias the category of crossed modules of dialgebras and morphisms of crossed modules. Just like for groups, it is possible to define the full embeddings J0 0:Dias →XDias and J0 1:Dias →XDias, with J0 0(D) = ({0}, D, 0) and J0 1(D) = (D, D, idD) for any dialgebra D. Given a morphism of dialgebras α:D→D0,J0 0(α) = (0, α) and J0 1(α) = (α, α). Besides, let us define the functors Ψ0 0,Ψ0 1and Ψ0 2, from XDias to Dias, given by Ψ0 0(L, D, µ) = D/µ(L), Ψ0 1(L, D, µ) = Dand Ψ0 2(L, D, µ) = Lfor any crossed module of dialgebras (L, D, µ). Given a morphism of crossed modules of dialgebras (ϕ, ψ): (L, D, µ)→(L0, D0, µ0), Ψ0 0(ϕ, ψ) = ψ,Ψ0 1(ϕ, ψ) = ψand Ψ0 2(ϕ, ψ) = ϕ, where ψis the morphism from D/µ(L) to D0/µ0(L0) induced by ψ. Proposition 1.2.57. Ψ0 0is left adjoint to J0 0,J0 0is left adjoint to Ψ0 1,Ψ0 1is left adjoint to J0 1and J0 1is left adjoint to Ψ0 2. Proof. The corresponding natural bijections can be readily described by using Example 1.2.56 (i)–(iv). 48 1 Crossed modules and equivalent structures Leibniz 2-algebras are defined as a particular case of semistrict Leibniz 2-algebras. In order to define this more flexible structure, it is necessary to endow 2Mod with a 2-category structure, i.e. it is mandatory to consider internal natural transformations. By doing so, it is possible to define a trilinear natural isomorphism, the Jacobiator, along with the Jacobiator identity, which relates the two ways the Jacobiator can be used in order to rebracket [[[x, y], z], w] for x, y, z, w ∈L0. The Jacobiator weakens the Leibniz condition (1.3.1), making it into a natural isomorphism instead of an identity. Moreover, in the semistrict framework, the identities 1.3.2 in the definition of a Leibniz 2-algebra homomorphism are replaced by a bilinear natural transformation. Definition 1.3.10. A strict 2-dialgebra is a 2-module D, together with two bilinear functors, the left and the right products −a−:D×D→Dand −`−:D×D→D, such that (xaiy)aiz=xai(y`iz),(1.3.3) (xaiy)aiz=xai(yaiz),(1.3.4) (x`iy)aiz=x`i(yaiz),(1.3.5) (xaiy)`iz=x`i(y`iz),(1.3.6) (x`iy)`iz=x`i(y`iz).(1.3.7) for all x, y, z ∈Di,i= 0,1. A morphism of strict 2-dialgebras, F:D→D0, is a linear functor such that Fi(xaiy) = Fi(x)a0 iFi(y)and Fi(x`iy) = Fi(x)`0 iFi(y) (1.3.8) for all x, y ∈Di,i= 0,1. We will denote by S2Dias the category of strict 2-dialgebras and the corresponding homomorphisms. The notion of semistrict 2-dialgebra has not been explored as far as we now. The construction followed in the case of semistrict Leibniz 2-algebras suggests that it would be necessary to weaken (1.3.3)–(1.3.7) by introducing five trilinear natural isomorphisms, the associators, and the corresponding coherence laws in order to relate all the possible ways of using the associators to rebracket all the possible combinations involving four elements in D0and the bilinear functors `and a. However, this approach is just a thought and it has not yet been proved to be valid. It is our intention to consider semistrict 2-dialgebras as a possible aim for further research. Observe that the difference between the previous definitions and the corresponding internal categories is merely semantic. Let us show it for the dialgebra case. In the definition of an internal category D= (D1, D0, s, e, t, κ) in Dias, the dialgebra structure of Diis precisely given by two bilinear maps, − ai−:Di×Di→Diand − `i−:Di×Di→Di, such that (1.3.3)–(1.3.7) are satisfied, for i= 0,1. On the other hand, the fact that s,t,eand κpreserve the dialgebra structure is equivalent 1.3 Categorification of algebraic structures and crossed modules 49 to the commutativity of the diagrams D1⊕D1D0⊕D0D1⊕D1D0⊕D0D1⊕D1D0⊕D0 D1D0D1D0D1D0 s⊕s a1a0 t⊕t a1a0a1 e⊕e a0 ste (D1⊕D1)×D0⊕D0(D1⊕D1)D1×D0D1 D1⊕D1D1 (a1,a1) κ⊕κκ a1 along with the analogous versions involving `0and `1. As for morphisms, in the definition of an internal functor Fbetween two internal categories in Dias, it is obvious that condition (1.3.8) is equivalent to F0and F1being morphisms of dialgebras. This situation is completely analogous in the Leibniz case. Hence, we have the following theorem: Theorem 1.3.11. The categories IDias (respectively ILb) and S2Dias (respectively S2Lb) are isomorphic. Bearing in mind Theorem 1.1.20 we have: Corollary 1.3.12. The categories XDias (respectively XLb) and S2Dias (respectively S2Lb) are equivalent. 50 1 Crossed modules and equivalent structures Chapter 2 Actors and modules over crossed modules In Section 2.1 we recall the construction of the algebra of bimultipliers and the Leibniz algebra of biderivations, which are, under certain conditions, the actor in the categories As and Lb respectively. Additionally we construct the dialgebra of tetramultipliers and prove that it is the actor in Dias under certain similar conditions. In Section 2.2 we recall the construction of the actor crossed module in XGr and XLie and give the description of a general actor in XLb that becomes the actor under certain conditions. In Section 2.3 we define the notion of left module over crossed modules of Lie algebras and recall the definitions of left modules over crossed modules of associative algebras and groups. 2.1 Actors in categories of interest Given a group H, we denote by Aut(H) the group of automorphisms of H. The morphism α:H→Aut(H), where α(h)(h0) = hh0h−1, together with the action of Aut(H) on Hdefined by ϕh=ϕ(h) for all ϕ∈Aut(H), h∈His a crossed module. Furthermore, for every action of a group Gon Hthere is a unique group homomorphism β:G→Aut(H) with gh=β(g)h. Conversely, every group homomorphism from Gto Aut(H) induces an action of Gon H. Therefore it would be possible to define a group action of Gon Has a group homomorphism from Gto Aut(H). The analogue to automorphisms of groups for a Lie algebra mis Der(m), the Lie algebra of derivations of m. Recall that an element in Der(m) is a K-linear map dfrom mto msuch that d([m1, m2]) = [d(m1), m2]+[m1, d(m2)] for all m1, m2∈m. The Lie structure is given by the bracket [d1, d2] = d1d2−d2d1for all d1, d2∈Der(m). The Lie homomorphism α:m→Der(m), where α(m)(m0) = [m, m0], is a crossed module 51 52 2 Actors and modules over crossed modules together with the action of Der(m) on mdefined by [ϕ, m] = ϕ(m) for all ϕ∈Der(m), m∈m. Just like for groups, given an action of a Lie algebra pon a Lie algebra mthere is a unique morphism of Lie algebras β:p→Der(m), such that [p, m] = [β(p), m]. Bearing this in mind, Casas, Datuashvili and Ladra give the following definition in [19] (see also [11]). Definition 2.1.1 ([19]).Let Cbe a category of interest. For any object Ain C, an actor of Ais a crossed module (A, Act(A), α)such that for any object C∈ C and an action of Con A, there is a unique morphism β:C→Act(A)with ca=β(c)a, c∗a=β(c)∗afor any a∈A,c∈C,∗ ∈ Ω0 2. It follows immediately from the previous definition that, given an object Ain C, an actor Act(A) is a unique object up to isomorphism. In [19, Definition 3.9] and the later paper [20, Proposition and Definition 3.1], Casas, Datuashvili and Ladra give an equivalent definition of the actor, in which the condition of the existence of a crossed module (A, Act(A), α) is changed by simply asking for Act(A) to have a set of derived actions on A. In fact, given an object A in C, there is always a set of derived actions of Aon itself, given by conjugation for the group operation and simply by the operations themselves for Ω0 2. If the actor exists, there is a unique morphism β:A→Act(A), which is a crossed module in C[20, Proposition 3.5 (a)]. This “upgraded” definition is equivalent to that of split extension classifier from [10]. It is also proved in [19, 20] that the actor in the case of associative algebras and Leibniz algebras does not exist unless some extra conditions are considered. Let us briefly recall the distinctive features of those particular cases. The following definition is closely related to the notion of multiplication of a ring by Hochschild [53], called bimultiplication by Mac Lane [68]. Definition 2.1.2. Let Bbe an associative algebra. A bimultiplier of Bis a pair (l, r) of K-linear maps l, r:B→Bsuch that l(b·b0) = l(b)·b0, r(b·b0) = b·r(b0), b·l(b0) = r(b)·b0. for all b, b0∈B. We will denote by Bim(B) the set of bimultipliers of B. Observe that these three conditions are a perfect match with the identities (1), (2) and (3) from Definition 1.2.24. It is clear that, given an element b∈B, the pair (lb, rb), with lb(b0) = b·b0and rb(b0) = b0·bfor all b0∈B, is a bimultiplier. The K-module structure of Bim(B) is obvious and its algebra structure is given by: (l1, r1)·(l2, r2) = (l1l2, r2r1) 2.1 Actors in categories of interest 53 for all (l1, r1),(l2, r2)∈Bim(B). It is immediate to check that the map α:B→Bim(B), b7→ α(b) = (lb, rb), with lband rbas defined previously, is a morphism of algebras. The problem is that the set of actions of Bim(B) on Bdoes not satisfy all the axioms of what we call an action of an algebra, i.e. it is not a set of derived actions. That set of actions is given by (l, r)·b=l(b), b·(l, r) = r(b), for all (l, r)∈Bim(B), b∈B. All the axioms from Definition 1.2.24 are satisfied, except number (5). Let (l1, r1),(l2, r2)∈Bim(B) and b∈B. Then ((l1, r1)·b)·(l2, r2) = l1(b)·(l2, r2) = r2(l1(b)),(2.1.1) but (l1, r1)·(b·(l2, r2)) = (l1, r1)·r2(b) = l1(r2(b)).(2.1.2) In general r2l1(b) and l1r2(b) are not necessarily equal. Recall that a category of interest is a category of groups with operations with two additional axioms (see Definition 1.1.4). In [19, 20], given a category of interest C with the set of identities E, it is denoted by EGthe subset of E that includes all the identities except those from the additional axioms satisfied by a category of interest. The category with the same set of operations and EGas the set of identities is denoted by CG. It is immediate that there is a full inclusion functor C,→ CG. The set of actions of Bim(B) on Bis indeed a set of derived actions in AsG, since it satisfies the conditions from Lemma 1.1.8, that is bilinearity, since the group action is trivial due to the commutativity of the addition. Moreover, (B, Bim(B), α) is a crossed module in AsGand given an action of an algebra Aon B, there is a morphism of algebras β:A→Bim(B) such that a·b=β(a)·b, for any a∈A,b∈B. In other words, Bim(B) is what Casas, Datuashvili and Ladra call a general actor of B: Definition 2.1.3 ([19, 20]).Let Cbe a category of interest and Aan object in C. A general actor object GAct(A)of Ais an object in CGthat has a set of actions on A, which is a set of derived actions in CG, such that for any object Cin Cwith a set of derived actions on Ain C, there exist a unique morphism in CG,β:C→GAct(A), with ca=β(c)a,c∗a=β(c)∗afor all c∈C,a∈A,∗ ∈ Ω0 2. Observe that, unlike the actor, a general actor is not necessarily unique. In fact, in [19], for an object Ain a category of interest C, it is constructed a general actor, denoted B(A) such that B(A),→Bim(A) when C=As. In order to construct B(A), Casas, Datuashvili and Ladra consider all the split extensions of Ain C, that is all the objects Bin Cwith a set of derived actions on Ain C. Then, for any element b in any of those objects Bthey consider the maps from Ato Awhich take an element ain Ato the result of bacting on a, one for every action in the set of actions of B. Afterwards, they define a set of operations for all those action maps and denote by 54 2 Actors and modules over crossed modules B(A) the set of all those maps and the new maps obtained as a result of operating the initial ones. In [20], due to a recommendation by Z. Janelidze, the authors included the notions of strict action [20, Definition 3.2] and strict general actor (which is a general actor whose action on the corresponding object is strict), along with a condition on a general actor [20, Condition A, p. 100] in order to define the universal strict general actor. Note that Bim(B) (and Bider(m), which is described below) is not necessarily a universal strict general actor, although it is a strict general actor. B(A) as constructed in [19] is in fact a universal strict general actor [20, Theorem 4.3], which is unique up to isomorphism if it is appropriate (in the sense of the equalities in Condition A) to the given presentation of the category of interest. Additionally, if an object Ahas an actor, then Act(A) = B(A) [20, Proposition 4.7]. Back to the particular situation of associative algebras, the major weakness of Bim(B) is that, in general, for any algebra Athere are morphisms from Ato Bim(B) that do not induce a set of derived actions of Aon B. Nevertheless, in [19, 20] the authors give a particular case of associative algebras for which the actor is indeed the algebra of bimultipliers, which follows directly from the following result. Lemma 2.1.4. Let Bbe an associative algebra such that Ann(B) = 0 or B2=B. Then, given (l1, r1),(l2, r2)∈Bim(B), r2l1(b) = l1r2(b). for any b∈B. Proof. Let us first assume that Ann(B) = 0. Let b, b0∈B. Then (l1r2(b)−r2l1(b)) ·b0=l1r2(b)·b0−r2l1(b)·b0=l1(r2(b)·b0)−l1(b)·l2(b0) =l1(b·l2(b0)) −l1(b)·l2(b0) = l1(b)·l2(b0)−l1(b)·l2(b0)=0, just by using the properties of the bimultipliers. By similar calculations, one can easily prove that b0·(l1r2(b)−r2l1(b)) = 0. Hence, l1r2(b)−r2l1(b) is an element of the annihilator and l1r2(b) = r2l1(b). If we consider B2=Bas the hypothesis, it would be sufficient to prove that l1r2(b·b0) = r2l1(b·b0) for any pair of elements b, b0∈B, but that identity follows almost immediately from the properties of the bimultipliers. Observe that the only impediment for Bim(B) to be the actor of Bwas that in general (2.1.1) and (2.1.2) are not equal. Hence, directly from the previous lemma, we have the following result. Proposition 2.1.5 ([19, 20]).Let Bbe an associative algebra such that Ann(B)=0 or B2=B. Then Act(B) = Bim(B). 2.1 Actors in categories of interest 55 Concerning Leibniz algebras, the situation is quite similar. Given a Leibniz algebra m, the role of general actor is played by the Leibniz algebra of biderivations Bider(m), described for the first time by Loday [64]. Definition 2.1.6. Let mbe a Leibniz algebra. A biderivation of mis a pair (d, D)of K-linear maps d, D:m→msuch that d([m, m0]) = [d(m), m0]+[m, d(m0)],(2.1.3) D([m, m0]) = [D(m), m0]−[D(m0), m],(2.1.4) [m, d(m0)] = [m, D(m0)],(2.1.5) for all m, m0∈m. In other words, dis a derivation (first identity) and Dis an anti-derivation (second identity), which additionally satisfy the third condition. It is not difficult to check that, given an element m∈m, the pair (ad(m),Ad(m)), with ad(m)(m0) = −[m0, m] and Ad(m)(m0)=[m, m0] for all m0∈m, is a biderivation. Loday calls (ad(m),Ad(m)) the inner biderivation of m. The K-module structure of Bider(m) is obvious and its Leibniz structure is given by [(d1, D1),(d2, D2)] = (d1d2−d2d1, D1d2−d2D1) (2.1.6) for all (d1, D1),(d2, D2)∈Bider(m). Checking that [(d1, D1),(d2, D2)] is indeed a biderivation of mis fairly simple, although the identity (2.1.5) is not completely straightforward. Nevertheless, it can be easily derived from the following result. Lemma 2.1.7. Let mbe a Leibniz algebra and (d1, D1),(d2, D2)∈Bider(m). Then [D1d2(m), m0] = [D1D2(m), m0], [m, D1d2(m0)] = [m, D1D2(m0)], for all m, m0∈m. Proof. Let m, m0∈mand (d1, D1),(d2, D2)∈Bider(m). According to the identity (2.1.5) for (d2, D2), [m0, d2(m)] = [m0, D2(m)], so D1([m0, d2(m)]) = D1([m0, D2(m)]). Simply by using the fact that D1is an anti-derivation we get that [D1(m0), d2(m)] −[D1d2(m), m0]=[D1(m0), D2(m)] −[D1D2(m), m0]. Therefore [D1d2(m), m0]=[D1D2(m), m0], since [D1(m0), d2(m)] = [D1(m0), D2(m)] due to (2.1.5) for (d2, D2). Analogously, d1([m, d2(m0)]) = d1([m, D2(m0)]). If we use the fact that d1is a derivation, [d1(m), d2(m0)] + [m, d1d2(m0)] = [d1(m), D2(m0)] + [m, d1D2(m0)]. Since [d1(m), d2(m0)] = [d1(m), D2(m0)], we have that [m, d1d2(m0)] = [m, d1D2(m0)]. Hence, [m, D1d2(m0)] = [m, D1D2(m0)] due to the identity (2.1.5) for (d1, D1). 56 2 Actors and modules over crossed modules It is easy to check that the map α:m→Bider(m), m7→ α(m) = (ad(m),Ad(m)) is a morphism of Leibniz algebras, but, analogously to what happens with the algebra of bimultipliers, there is a problem with the set of actions of Bider(m) on m(cf. [29]). That set of actions is given by [(d, D), m] = D(m), [m, (d, D)] = −d(m), for all (d, D)∈Bider(m), m∈m. All the axioms from Definition 1.2.37 are satisfied, except number (6). Let (d1, D1),(d2, D2)∈Bider(m) and m∈m. Then [(d1, D1),[(d2, D2), m]] = [(d1, D1), D2(m)] = D1(D2(m)), but [[(d1, D1),(d2, D2)], m]−[[(d1, D1), m],(d2, D2)] = [(d1d2−d2d1, D1d2−d2D1), m] −[D1(m),(d2, D2)] = D1(d2(m)) −d2(D1(m)) + d2(D1(m)) = D1(d2(m)). In general D1D2(m) and D1d2(m) are not necessarily equal. Similarly to the situation for associative algebras, there is a sufficient condition in order to guarantee that Bider(m) is the actor of m, which follows directly from Lemma 2.1.7. Proposition 2.1.8 ([19, 20]).Let mbe a Leibniz algebra such that Ann(m) = 0 or [m,m] = m. Then Act(m) = Bider(m). Proof. Let m∈mand (d1, D1),(d2, D2)∈Bider(m). Recall that the issue that prevents Bider(m) from being the actor of mis that D1D2(m) and D1d2(m) are not equal in general. Due to Lemma 2.1.7, D1d2(m)−D1D2(m)∈Ann(m). Hence, if we assume that Ann(m) = 0, D1d2(m) = D1D2(m). Let us now work under the hypothesis [m,m] = m. In this situation it is enough to prove that D1d2([m, m0]) = D1D2([m, m0]) for any pair of elements m, m0∈m. By applying (2.1.3) and (2.1.4) we get that D1D2([m, m0]) = [D1D2(m), m0]−[D1(m0), D2(m)] −[D1D2(m0), m]+[D1(m), D2(m0)], D1d2([m, m0]) = [D1d2(m), m0]−[D1(m0), d2(m)] + [D1(m), d2(m0)] −[D1d2(m0), m], so D1d2([m, m0]) = D1D2([m, m0]), due to Lemma 2.1.7 and the identity (2.1.5) for (d2, D2). In the next subsection we construct an object in Dias analogous to the algebra of bimultipliers in As and the Leibniz algebra of biderivations in Lie. 2.1.1 The actor in the category of dialgebras 57 2.1.1 The actor in the category of dialgebras In [19, 20] there is a description of a method to construct a general actor of an object in a category of interest. However, the dialgebra presented here is constructed by analogy to the algebra of bimultipliers. Definition 2.1.9. Let Lbe a dialgebra. We denote by Tetra(L)the set of tetramultipliers of L, whose elements are quadruples t= (l, r, ˜ l, ˜r)of K-linear maps from Lto Lsuch that (1) l(a`b) = l(a)ab, (2) l(aab) = l(a)ab, (3) ˜ l(aab) = ˜ l(a)ab, (4) ˜ l(a`b) = l(a)`b, (5) ˜ l(a`b) = ˜ l(a)`b, (6) r(a)ab=aa˜ l(b), (7) r(a)ab=aal(b), (8) ˜r(a)ab=a`l(b), (9) r(a)`b=a`˜ l(b), (10) ˜r(a)`b=a`˜ l(b), (11) r(aab) = aa˜r(b), (12) r(aab) = aar(b), (13) r(a`b) = a`r(b), (14) ˜r(aab) = a`˜r(b), (15) ˜r(a`b) = a`˜r(b), for all a, b ∈L. Note that the aim is to construct an object which can be used to describe every action on L. Therefore it makes sense to consider elements that respect the axioms of a dialgebra action (see conditions (1)–(15)) from Definition 1.2.50. Lemma 2.1.10. Let Lbe a dialgebra. Given a∈L, the quadruple (la, ra,˜ la,˜ra), with la(a0) = aaa0, ra(a0) = a0aa, ˜ la(a0) = a`a0,˜ra(a0) = a0`a, for all a0∈L, is an element in Tetra(L). Proof. Given a∈L, (la, ra,˜ la,˜ra) verifies conditions (1)–(15) from Definition 2.1.9 directly from the five di-associativity axioms in L(see Definition 1.2.47). Let us now define left and right products in Tetra(L). 64 2 Actors and modules over crossed modules for all h, h0∈H,p, p0∈P,m∈M,g∈G. Proof. Let us first suppose that (H, G, ∂) acts on (M, P, µ), that is there is a morphism of crossed modules H G D(P, M) Aut(µ) ∂ ϕψ ∆ (2.2.3) The previous diagram is commutative and ϕ(gh) = ψ(g)ϕ(h) for all h∈H,g∈G. We will denote ψ(g) by (σg, θg) for any g∈G. There is an action of Gon M (respectively P) given by gm=σg(m) (respectively gp=θg(p)) for all g∈G,m∈M (respectively p∈P), which induces an action of Hon M(respectively P) via ∂. The identities µ(gm) = gµ(m) and g(pm) = (gp)(gm) follow from µσg=θgµand σg(pm) = θg(p)σg(m) respectively, for all g∈G,m∈M,p∈P. Therefore (i) holds. Regarding (ii), we can define ξ(h, p) = ϕ(h)(p) for any h∈H,p∈P. In this way, (GrM1) and (GrM2) follow from the commutativity of (2.2.3). (GrM3) follows from the identity ϕ(gh) = ψ(g)ϕ(h) for all h∈H,g∈G, and the definition of the action of Aut(µ) on D(P, M). (GrM4) is an immediate consequence of ϕbeing a morphism of groups and the definition of ◦in D(P, M). Finally, (GrM5) is easy to prove by using that ϕ(h) is a derivation for all h∈H. The converse statement is rather obvious. If we assume that (i) and (ii) hold, it is possible to define a morphism of crossed modules (ϕ, ψ) from (H, G, ∂) to Act(M, P, µ) as follows. Given h∈H,ϕ(h)(p) = ξ(h, p) for all p∈P. For any g∈G,ψ(g) = (σg, θg), with σg(m) = gmand θg(p) = gpfor all m∈M,p∈P. Given h∈H, (GrM5) guarantees that ϕ(h) is a derivation. Furthermore, ϕ(h−1) is the inverse of ϕ(h) in D(P, M) and ϕis a group homomorphism due to (GrM2) and (GrM4). Concerning ψ, given g∈G,σgand θgare automorphisms as a direct consequence of the three identities satisfied by the group actions of Gon Mand Prespectively. Additionally (σg, θg) is a morphism of crossed modules due to the identities µ(gm) = gµ(m) and g(pm) = (gp)(gm) for all g∈G,m∈M,p∈P. The identity ∆ϕ=ψ∂ follows immediately from (GrM1) and (GrM2). Finally, the fact that ϕ(gh) = ψ(g)ϕ(h) for all h∈H,g∈Gis a consequence of (GrM3). Let (H, G, ∂) be a crossed module of groups acting on the crossed module (M, P, µ). By Proposition 2.2.2, Gacts on Pand Hacts on M, so it makes sense to consider the semidirect products of groups PoGand MoH. There is an action of PoGon MoH given by (p,g)(m, h) = (p(gm)(ξ(gh, p−1)),gh) for all (p, g)∈PoG, (m, h)∈MoH. The morphism of groups (µ, ∂): MoH→PoG, (m, h)7→ (µ(m), ∂(h)) is a crossed module together with that action. (MoH, P oG, (µ, ∂)) is called the semidirect product of the crossed modules (M, P, µ) and (H, G, ∂). Note that the semidirect product determines an obvious 2.2.2 Actor crossed module of Lie algebras 65 split extension of (H, G, ∂) by (M, P, µ) (0,0,0) (M, P, µ) (MoH, P oG, (µ, ∂)) (H, G, ∂) (0,0,0) Conversely, any split extension of (H, G, ∂) by (M, P, µ) is isomorphic to their semidirect product, where the action of (H, G, ∂) on (M, P, µ) is induced by the splitting morphism. Therefore, the definition of an action of a crossed module of groups on another crossed module of groups agrees with the general notion of derived action in a category of Ω-groups. Recall that XGr is indeed a category of interest (see [75]). 2.2.2 Actor crossed module of Lie algebras It is possible to construct the actor of a Lie crossed module (see [27]), following a similar procedure to the one described for groups. Let us recall that construction in order to appreciate the slight differences. Note that in [27] Kis considered a field, but the procedure still works for Ka commutative unital ring. Let (n,q, µ) be a Lie crossed module. Consider Der(q,n), the K-module of all derivations from qto n, that is all the K-linear maps d:q→nsuch that d([q, q0]) = [d(q), q0]+[q, d(q0)] (2.2.4) for all q, q0∈q. There is a Lie bracket in Der(q,n) given by [d1, d2] = d1µd2−d2µd1 for all d1, d2∈Der(q,n). Just like for groups, the Peiffer identity and the equivariance are essential ir order to prove that the result of this bracket lies within Der(q,n). The antisymmetry and the Jacobi identity follow directly from the definition of the bracket. If we consider the crossed module (q,q,idq), Der(q,q) is the Lie algebra of derivations of q, denoted simply by Der(q). The other Lie algebra required to define the actor crossed module is Der(n,q, µ), the Lie algebra of derivations of the crossed module (n,q, µ), whose elements are all pairs (σ, θ), with σ∈Der(n) and θ∈Der(q), such that θµ =µσ and σ([q, n]) = [q, σ(n)] + [θ(q), n] (2.2.5) for all n∈n,q∈q. The Lie structure of Der(n,q, µ) is given by (σ1, θ1)+(σ2, θ2)=(σ1+σ2, θ1+θ2), λ(σ, θ)=(λσ, λθ), [(σ1, θ1),(σ2, θ2)] = ([σ1, σ2],[θ1, θ2]), for all (σ1, θ1),(σ2, θ2),(σ, θ)∈Der(n,q, µ), λ∈K. Recall that the bracket in the Lie algebra of derivations of a Lie algebra is described just before Example 1.2.17 and in the last paragraph from page 51. 66 2 Actors and modules over crossed modules There is a Lie homomorphism ∆: Der(q,n)→Der(n,q, µ), given by ∆(d)=(dµ, µd) (2.2.6) for all d∈Der(q,n). Observe that the Peiffer identity of (n,q, µ) guarantees that dµ is a derivation of n, while µd is a derivation of qdue to the equivariance. As for the condition (2.2.5), the first identity is obvious and the second one: dµ([q, n]) = d([q, µ(n)]) = [d(q), µ(n)] + [q, dµ(n)] = [µd(q), n]+[q, dµ(n)], for all n∈n,q∈q, follows from the equivariance, (2.2.4) and the Peiffer identity. Furthermore, there is a Lie action of Der(n,q, µ) on Der(q,n) defined by [(σ, θ), d] = σd −dθ. (2.2.7) for all (σ, θ)∈Der(n,q, µ), d∈Der(q,n). It is a matter of routine calculations to check that σd −dθ is indeed a derivation from qto nand the identities [[(σ1, θ1),(σ2, θ2)], d] = [(σ1, θ1),[(σ2, θ2), d]] −[(σ2, θ2),[(σ1, θ1), d]], [(σ, θ),[d1, d2]] = [[(σ, θ), d1], d2]+[d1,[(σ, θ), d2]] hold for all (σ, θ),(σ1, θ1),(σ2, θ2)∈Der(n,q, µ), d, d1, d2∈Der(q,n). Moreover, we have that ∆([(σ, θ), d]) = ∆(σd −dθ) = ((σd −dθ)µ, µ(σd −dθ)) = (σdµ −dθµ, µσd −µdθ)=(σdµ −dµσ, θµd −µdθ) = [(σ, θ),(dµ, µd)] = [(σ, θ),∆(d)], [∆(d1), d2] = [(d1µ, µd1), d2] = d1µd2−d2µd1= [d1, d2], so (Der(q,n),Der(n,q, µ),∆) is a Lie crossed module, called the actor crossed module of (n,q, µ) and denoted by Act(n,q, µ). Definition 2.2.3 ([27]).An action of a Lie crossed module (m,p, ν)on another Lie crossed module (n,q, µ)is a morphism of Lie crossed modules from (m,p, ν)to Act(n,q, µ). Example 2.2.4. There is an action of a Lie crossed module (m,p, ν)on itself given by the morphism (ϕν, ψν): (m,p, ν)→Act(m,p, ν), where ϕν(m)(p) = −[p, m]and ψν(p) = (σp, θp)with σp(m) = [p, m]and θp(p0) = [p, p0]for all m∈m,p, p0∈p(see [27] for more details). By analogy to Proposition 2.2.2, we give an equivalent description of an action of a Lie crossed module on another Lie crossed module in terms of equations. 2.2.2 Actor crossed module of Lie algebras 67 Proposition 2.2.5. Let (m,p, ν)and (n,q, µ)be Lie crossed modules. There is an action of (m,p, ν)on (n,q, µ)if and only if the following conditions hold: (i) There are actions of the Lie algebra p(and so m) on the Lie algebras nand q; µis a p-equivariant homomorphism, that is µ([p, n]) = [p, µ(n)] (LieEQ) and the actions of pand qon nare compatible, that is [[p, q], n] = [p, [q, n]] −[q, [p, n]] (LieCOM) for all p∈p,q∈qand n∈n. (ii) There is a K-bilinear map ξ:m×q→nsuch that µξ(m, q)=[m, q],(LieM1) ξm, µ(n)= [m, n],(LieM2) [p, ξ(m, q)] = ξ([p, m], q) + ξ(m, [p, q]),(LieM3) ξ([m, m0], q)=[m, ξ(m0, q)] −[m0, ξ(m, q)],(LieM4) ξ(m, [q, q0]) = [q, ξ(m, q0)] −[q0, ξ(m, q)],(LieM5) for all m, m0∈m,q, q0∈q,n∈n,p∈p. Proof. Let us first assume that (m,p, ν) acts on (n,q, µ), that is there is a morphism of crossed modules m p Der(q,n) Der(n,q, µ) ν ϕψ ∆ (2.2.8) Given p∈p, let us denote ψ(p) by (σp, θp), with σp∈Der(n), θp∈Der(q) such that θpµ=µσpand σp([q, n]) = [q, σp(n)] + [θp(q), n].(2.2.9) Due to (2.2.6), the commutativity of (2.2.8) can be expressed by the identity (ϕ(m)µ, µϕ(m)) = (σν(m), θν(m)),(2.2.10) for all m∈m. There is an action of pon n(respectively q) given by [p, n] = σp(n) (respectively [p, q] = θp(q)) for all p∈p,n∈n(respectively q∈q), which induces an action of mon n(respectively q) via ν. (LieEQ) and (LieCOM) follow from the first and the second identity in (2.2.9) respectively. Therefore (i) holds. 68 2 Actors and modules over crossed modules Concerning (ii), we can define ξ(m, q) = ϕ(m)(q) for all m∈m,q∈q. Let us show that (LieM1), (LieM2) and (LieM4) follow from (2.2.10). Given m, m0∈m,q∈qand n∈n, µξ(m, n) = µ(ϕ(m)(q)) = θν(m)(q)=[ν(m), q] = [m, q], ξ(m, µ(n)) = ϕ(m)(µ(n)) = σν(m)(n)=[ν(m), n] = [m, n], ξ([m, m0], q) = ϕ([m, m0])(q)=[ϕ(m), ϕ(m0)](q) = (ϕ(m)µϕ(m0)−ϕ(m0)µϕ(m))(q) =σν(m)(ϕ(m0)(q)) −σν(m0)(ϕ(m)(q)) = [m, ξ(m0, q)] −[m0, ξ(m, q)]. Regarding (LieM3), since (ϕ, ψ) is a morphism of Lie crossed modules, we know that ϕ([p, m]) = [(σp, θp), ϕ(m)]. Hence, ξ([p, m], q) + ξ(m, [p, q]) = [(σp, θp), ϕ(m)](q) + ϕ(m)(θp(q)) =σpϕ(m)(q)−ϕ(m)θp(q) + ϕ(m)θp(q)=[p, ξ(m, q)], for any m∈m,p∈p,q∈q. Finally, (LieM5) follows easily from the fact that ϕ(m) is a derivation from qto nfor any m∈m. Now, let us prove the converse statement. From (i), pacts on nand q, that is there are two bilinear maps p×n→n, (p, n)7→ [p, n] and p×q→q, (p, q)7→ [p, q] such that [[p, p0], n] = [p, [p0, n]] −[p0,[p, n]] ,(2.2.11) [p, [n, n0]] = [[p, n], n0]+[n, [p, n0]] ,(2.2.12) and [[p, p0], q]=[p, [p0, q]] −[p0,[p, q]] ,(2.2.13) [p, [q, q0]] = [[p, q], q0]+[q, [p, q0]] ,(2.2.14) for all n, n0∈n,p, p0∈p,q, q0∈q. It is possible to define a morphism of crossed modules (ϕ, ψ) from (m,p, ν) to Act(n,q, µ) as follows. Given m∈m,ϕ(m)(q) = ξ(m, q) for all q∈q. For any p∈p,ψ(p)=(σp, θp), with σp(n) = [p, n] and θp(q)=[p, q] for all n∈n,q∈q. It follows directly from (LieM5) that ϕ(m) is a derivation from qto nfor all m∈m. Moreover, given m, m0∈mand q∈q, [ϕ(m), ϕ(m0)](q) = ϕ(m)µϕ(m0)(q)−ϕ(m0)µϕ(m)(q) =ξ(m, µξ(m0, q)) −ξ(m0, µξ(m, q)) = [m, ξ(m0, q)] −[m0, ξ(m, q)] =ξ([m, m0], q) = ϕ([m, m0])(q), 2.2.2 Actor crossed module of Lie algebras 69 due to (LieM2) and (LieM4). Hence, ϕis a morphism of Lie algebras. As for ψ, given p∈p,σp(respectively θp) is a derivation of n(respectively q) due to (2.2.12) (respectively (2.2.14)). (LieEQ) and (LieCOM) guarantee that the pair (σp, θp) satisfies the identities (2.2.9). Also, it can be readily checked that ψis a Lie homomorphism by using (2.2.11) and (2.2.13). Recall that ∆ϕ(m)=(ϕ(m)µ, µϕ(m)) and ψν(m)=(σν(m), θν(m)) for any m∈m. By making use of (LieM1), (LieM2) and the fact that macts on nand qvia ν, ϕ(m)µ(n) = ξ(m, µ(n)) = [m, n]=[ν(m), n] = σν(m)(n), µϕ(m)(q) = µξ(m, q)=[m, q]=[ν(m), q] = θν(m)(q), for all n∈n,q∈q. Therefore, ∆ϕ=ψν. Furthermore, given m∈mand p∈p, due to (2.2.7), [ψ(p), ϕ(m)] = [(σp, θp), ϕ(m)] = σpϕ(m)−ϕ(m)θp. On the other hand, by using (LieM3), we get that ϕ([p, m])(q) = ξ([p, m], q)=[p, ξ(m, q)] −ξ(m, [p, q]) = (σpϕ(m)−ϕ(m)θp)(q), for any q∈q. Hence, ϕ([p, m]) = [ψ(p), ϕ(m)] for all m∈m,p∈p. Let (m,p, ν) be a Lie crossed module acting on a Lie crossed module (n,q, µ). By Proposition 2.2.5 there are Lie actions of mon nand of pon q, so it makes sense to consider the semidirect products of Lie algebras nomand qop. Furthermore, we have the following result. Lemma 2.2.6. There is an action of the Lie algebra qopon the Lie algebra nom, given by [(q, p),(n, m)] = ([q, n]+[p, n]−ξ(m, q),[p, m]) (2.2.15) for all (q, p)∈qop,(n, m)∈nom, with ξas in Proposition 2.2.5. Moreover, the Lie homomorphism (µ, ν): nom→qop, given by (µ, ν)(n, m) = µ(n), ν(m) for all (n, m)∈nom, is a Lie crossed module together with the previous action. Proof. Firstly, it is necessary to prove that (2.2.15) describes a Lie action, that is the identities [[(q, p),(q0, p0)] ,(n, m)] = [(q, p),[(q0, p0),(n, m)]] −[(q0, p0),[(q, p),(n, m)]] , [(q, p),[(n, m),(n0, m0)]] = [[(q, p),(n, m)] ,(n0, m0)] + [(n, m),[(q, p),(n0, m0)]] , for all (q, p),(q0, p0)∈qop, (n, m),(n0, m0)∈nom. It is easy to check the first identity by making use of (LieCOM), (LieM3), (LieM5) and the analogous identity 70 2 Actors and modules over crossed modules for the actions of pon nand qon n. Regarding the second identity, straightforward calculations give rise to [(q, p),[(n, m),(n0, m0)]] = ([q, [n, n0]] | {z } (1) +[q, [m, n0]] | {z } (2) −[q, [m0, n]] | {z } (3) +[p, [n, n0]] | {z } (4) +[p, [m, n0]] | {z } (5) −[p, [m0, n]] | {z } (6) −ξ([m, m0], q) | {z } (7) ,[p, [m, m0]] | {z } (8) ), [[(q, p),(n, m)],(n0, m0)] = ([[q, n], n0] | {z } (10) −[ξ(m, q), n0] | {z } (20) −[m0,[q, n]] | {z } (30) +[[p, n], n0] | {z } (40) +[[p, m], n0] | {z } (50) −[m0,[p, n]] | {z } (60) +[m0, ξ(m, q)] | {z } (70) ,[[p, m], m0] | {z } (80) ), [(n, m),[(q, p),(n0, m0)]] = ([n, [q, n0]] | {z } (100 ) +[m, [q, n0]] | {z } (200 ) −[n, ξ(m0, q)] | {z } (300 ) +[n, [p, n0]] | {z } (400 ) +[m, [p, n0]] | {z } (500 ) −[[p, m0], n] | {z } (600 ) −[m, ξ(m0, q)] | {z } (700 ) ,[m, [p, m0]] | {z } (800 ) ). It follows easily that (i)=(i0)+(i00) for i= 1,4,8, due to the action of qon nand the actions of pon nand m. For i= 7, the identity follows from (LieM4). For i= 2,3 it is important to note that [q, [m, n0]] = [q, ξ(m, µ(n0))] = ξ(m, [q, µ(n0)]) + [µ(n0), ξ(m, q)] =ξ(m, µ([q, n0])) + [n0, ξ(m, q)] = [m, [q, n0]] −[ξ(m, q), n0], immediately from (LieM2), (LieM5), the antisymmetry of the bracket in nand the equivariance and the Peiffer identity of (n,q, µ). Finally, for i= 5,6, bearing in mind that the action of mon nis induced by the action of pon nvia ν, we have that [p, [m, n0]] = [p, [ν(m), n0]] = [[p, ν(m)], n0]+[ν(m),[p, n0]] = [ν([p, m]), n0]+[m, [p, n0]] = [[p, m], n0]+[m, [p, n0]], directly from the equivariance of (m,p, ν). Checking that (µ, ν) is a Lie homomorphism that satisfies equivariance and the Peiffer identity is also a matter of routine calculations. The Lie crossed module nom,qop,(µ, ν)is called the semidirect product of the Lie crossed modules (n,q, µ) and (m,p, ν). Note that the semidirect product determines an obvious split extension of (m,p, ν) by (n,q, µ) (0,0,0) (n,q, µ)nom,qop,(µ, ν)(m,p, ν) (0,0,0) 2.2.3 Actor crossed module of Leibniz algebras 71 Conversely, any split extension of (m,p, ν) by (n,q, µ) is isomorphic to their semidirect product, where the action of (m,p, ν) on (n,q, µ) is induced by the splitting morphism (see [26]). 2.2.3 Actor crossed module of Leibniz algebras In the previous two subsections we have seen that the actor in the categories of groups and Lie algebras has its corresponding 2-dimensional analogue. Given a Leibniz algebra m, Bider(m) is not necessarily the actor of m, since the set of actions of Bider(m) on mis not a set of derived actions in general. In fact, the actor of a Leibniz algebra does not always exist. Nevertheless, under certain conditions, Bider mis indeed the actor of m. In this section we will show that under similar conditions, the actor crossed module can also be constructed. Let us assume for the rest of this subsection that (n,q, µ) is a Leibniz crossed module. Definition 2.2.7. The set of biderivations from qto n, denoted by Bider(q,n), consists of all the pairs (d, D)of K-linear maps, d, D:q→n, such that d([q, q0]) = [d(q), q0]+[q, d(q0)],(2.2.16) D([q, q0]) = [D(q), q0]−[D(q0), q],(2.2.17) [q, d(q0)] = [q, D(q0)],(2.2.18) for all q, q0∈q. Analogously to the Lie case, we translate the notion of a biderivation of a Leibniz algebra into a biderivation between two Leibniz algebras via the action. Given n∈n, the pair of K-linear maps (ad(n),Ad(n)), where ad(n)(q) = −[q, n] and Ad(n)(q) = [n, q] for all q∈q, is clearly a biderivation from qto n, so Bider(q,n) is not an empty set. Observe that if we consider the crossed module (q,q,idq), Bider(q,q) is exactly the set of biderivations of q. Directly from the definition and the fact that (n,q, µ) is a Leibniz crossed module, we get the following result. Lemma 2.2.8. Let (d, D)∈Bider(q,n). Then (dµ, Dµ)∈Bider(n)and (µd, µD)∈ Bider(q). Proof. Let us show that (dµ, Dµ)∈Bider(n). It is obvious that dµ and Dµ are K-linear maps from nto n. Furthermore, given n, n0∈n, dµ([n, n0]) = d([µ(n), µ(n0)]) = [dµ(n), µ(n0)] + [µ(n), dµ(n0)] = [dµ(n), n0]+[n, dµ(n0)], Dµ([n, n0]) = D([µ(n), µ(n0)]) = [Dµ(n), µ(n0)] −[Dµ(n0), µ(n)] = [Dµ(n), n0]−[Dµ(n0), n], [n, dµ(n0)] = [µ(n), dµ(n0)] = [µ(n), Dµ(n0)] = [n, Dµ(n0)], 72 2 Actors and modules over crossed modules as a straightforward consequence of the Peiffer identity, the properties of the biderivation (d, D) and the fact that µis morphism of Leibniz algebras. In order to prove that (µd, µD)∈Bider(q) it is necessary to make use of the equivariance of (n,q, µ) instead of the Peiffer identity, but the procedure is quite similar. Also from Definition 2.2.7 we get the following result. Lemma 2.2.9. Let (d1, D1),(d2, D2)∈Bider(q,n). Then [D1µd2(q), q0] = [D1µD2(q), q0], [q, D1µd2(q0)] = [q, D1µD2(q0)], for all q, q0∈q. Proof. Let q, q0∈qand (d1, D1),(d2, D2)∈Bider(q,n). According to the identity (2.2.18) for (d2, D2), [q0, d2(q)] = [q0, D2(q)], so D1µ([q0, d2(q)]) = D1µ([q0, D2(q)]). Due to (2.2.17) and the equivariance of (q,n, µ), one can easily derive that [D1(q0), µd2(q)] −[D1µd2(q), q0]=[D1(q0), µD2(q)] −[D1µD2(q), q0]. By the Peiffer identity and (2.2.18) for (d2, D2), [D1(q0), µd2(q)] = [D1(q0), µD2(q)]. Therefore [D1µd2(q), q0]=[D1µD2(q), q0]. Analogously, d1µ([q, d2(q0)]) = d1µ([q, D2(q0)]). By the equivariance and (2.2.16) we get that [d1(q), µd2(q0)] + [q, d1µd2(q0)] = [d1(q), µD2(q0)] + [q, d1µD2(q0)]. Due to the Peiffer identity and (2.2.18) for (d2, D2), [d1(q), µd2(q0)] = [d1(q), µD2(q0)], so [q, d1µd2(q0)] = [q, d1µD2(q0)] and using (2.2.18) again, this time for (d1, D1), we get that [q, D1µd2(q0)] = [q, D1µD2(q0)]. Observe that Lemma 2.1.7 follows directly from the previous lemma for the particular case of the crossed module (m,m,idm). Bider(q,n) has an obvious K-module structure. Regarding its Leibniz structure, it is described in the next proposition. Proposition 2.2.10. Bider(q,n)is a Leibniz algebra with the bracket given by [(d1, D1),(d2, D2)] = (d1µd2−d2µd1, D1µd2−d2µD1) (2.2.19) for all (d1, D1),(d2, D2)∈Bider(q,n). Proof. Let (d1, D1),(d2, D2)∈Bider(q,n). In the first place, we have to confirm that [(d1, D1),(d2, D2)] is a biderivation from qto n. Linearity is obvious. The identities (2.2.16) and (2.2.17) for [(d1, D1),(d2, D2)] follow easily from the same identities for (d1, D1) and (d2, D2) along with the equivariance and the Peiffer identity of (n,q, µ). Concerning (2.2.18), it is an immediate consequence of the second identity in Lemma 2.2.9 together with (2.2.18) for both (d1, D1) and (d2, D2). Checking the Leibniz identity is just a matter of routine calculations. 2.2.3 Actor crossed module of Leibniz algebras 73 As an analogue to the Lie algebra of derivations of a crossed module, we state the following definition. Definition 2.2.11. The set of biderivations of the Leibniz crossed module (n,q, µ), denoted by Bider(n,q, µ), consists of all the quadruples ((σ1, θ1),(σ2, θ2)) such that (σ1, θ1)∈Bider(n)and (σ2, θ2)∈Bider(q),(2.2.20) µσ1=σ2µand µθ1=θ2µ, (2.2.21) σ1([q, n]) = [σ2(q), n]+[q, σ1(n)],(2.2.22) σ1([n, q]) = [σ1(n), q]+[n, σ2(q)],(2.2.23) θ1([q, n]) = [θ2(q), n]−[θ1(n), q],(2.2.24) θ1([n, q]) = [θ1(n), q]−[θ2(q), n],(2.2.25) [q, σ1(n)] = [q, θ1(n)],(2.2.26) [n, σ2(q)] = [n, θ2(q)],(2.2.27) for all n∈n,q∈q. Note that (2.2.22)–(2.2.27) are very similar to (2.2.16)–(2.2.18), but here the two maps that define the action of qon nhave to be considered, which is the reason why the identities appear duplicated. Given q∈q, it can be readily checked that ((σq 1, θq 1),(σq 2, θq 2)), where σq 1(n) = −[n, q], θq 1(n)=[q, n], σq 2(q0) = −[q0, q], θq 2(q0)=[q, q0], is a biderivation of the crossed module (n,q, µ). The following result is similar to Lemmas 2.1.7 and 2.2.9, but it combines elements in Bider(q,n) and Bider(n,q, µ). Lemma 2.2.12. Let ((σ1, θ1),(σ2, θ2)),((σ0 1, θ0 1),(σ0 2, θ0 2)) ∈Bider(n,q, µ)and (d, D)∈ Bider(q,n). Then [Dσ2(q), q0]=[Dθ2(q), q0], [q, Dσ2(q0)] = [q, Dθ2(q0)], [θ1d(q), q0] = [θ1D(q), q0], [q, θ1d(q0)] = [q, θ1D(q0)], [Dσ2(q), n]=[Dθ2(q), n], [n, Dσ2(q)] = [n, Dθ2(q)], [θ1d(q), n]=[θ1D(q), n], [n, θ1d(q)] = [n, θ1D(q)], [θ1σ0 1(n), q]=[θ1θ0 1(n), q], [q, θ1σ0 1(n)] = [q, θ1θ0 1(n)], [θ2σ0 2(q), n]=[θ2θ0 2(q), n], [n, θ2σ0 2(q)] = [n, θ2θ0 2(q)], for all n∈n,q, q0∈q. Proof. Let us begin with the first column. Let q, q0∈q, (d, D)∈Bider(q,n) and ((σ1, θ1),(σ2, θ2)) ∈Bider(n,q, µ). Since (σ2, θ2) is a biderivation of q, we have that [q0, σ2(q)] = [q0, θ2(q)]. Therefore D([q0, σ2(q)]) = D([q0, θ2(q)]). Directly from (2.2.17), we get that [D(q0), σ2(q)] −[Dσ2(q), q0] = [D(q0), θ2(q)] −[Dθ2(q), q0]. 80 2 Actors and modules over crossed modules and [p, [q, q0]] = [[p, q], q0]−[[p, q0], q],(2.2.42) [q, [p, q0]] = [[q, p], q0]−[[q, q0], p],(2.2.43) [q, [q0, p]] = [[q, q0], p]−[[q, p], q0],(2.2.44) [q, [p, p0]] = [[q, p], p0]−[[q, p0], p],(2.2.45) [p, [q, p0]] = [[p, q], p0]−[[p, p0], q],(2.2.46) [p, [p0, q]] = [[p, p0], q]−[[p, q], p0],(2.2.47) for all n, n0∈n,p, p0∈p,q, q0∈q. Recall that directly from (2.2.37), (2.2.38), (2.2.40) and (2.2.41), [n, [p, n0]] = −[n, [n0, p]],(2.2.48) [p, [n, p0]] = −[p, [p0, n]],(2.2.49) for all n, n0∈n,p, p0∈p. Analogously, [q, [p, q0]] = −[q, [q0, p]],(2.2.50) [p, [q, p0]] = −[p, [p0, q]],(2.2.51) for all q, q0∈q,p, p0∈p. It is possible to define a morphism of crossed modules (ϕ, ψ) from (m,p, η) to Act(n,q, µ) as follows. Given m∈m,ϕ(m)=(dm, Dm), with dm(q) = −ξ2(q, m), Dm(q) = ξ1(m, q), for all q∈q. On the other hand, for any p∈p,ψ(p) = ((σp 1, θp 1),(σp 2, θp 2)), σp 1(n) = −[n, p], θp 1(n)=[p, n], σp 2(q) = −[q, p], θp 2(q)=[p, q], for all n∈n,q∈q. It follows directly from (LbM5a)–(LbM5c) that (dm, Dm)∈ Bider(q,n) for all m∈m. Besides, ϕis clearly K-linear and given m, m0∈m, [ϕ(m), ϕ(m0)] = [(dm, Dm),(dm0, Dm0)] = [dmµdm0−dm0µdm, Dmµdm0−dm0µDm]. For any q∈q, dmµdm0(q)−dm0µdm(q) = −ξ2(µdm0(q), m) + ξ2(µdm(q), m0) =−[dm0(q), m]+[dm(q), m0] = [ξ2(q, m0), m]−[ξ2(q, m), m0] =−ξ2(q, [m, m0]) = d[m,m0](q), 2.2.3 Actor crossed module of Leibniz algebras 81 due to (LbM2a) and (LbM4a). Analogously, it can be easily checked the identity (Dmµdm0−dm0µDm)(q) = D[m,m0](q) by making use of (LbM2a), (LbM2b) and (LbM4b). Hence, ϕis a morphism of Leibniz algebras. As for ψ, it is necessary to prove that ((σp 1, θp 1),(σp 2, θp 2)) satisfies all the axioms from Definition 2.2.11 for any p∈p. The fact that (σp 1, θp 1) (respectively (σp 2, θp 2)) is a biderivation of n(respectively q) follows directly from (2.2.36), (2.2.38) and (2.2.48) (respectively (2.2.42), (2.2.44) and (2.2.50)). The identities µθp 1=θp 2µand µσp 1=σp 2µ are an immediate consequence of (LbEQ1) and (LbEQ2) respectively. Observe that the combinations of the identities (LbCOM1) and (LbCOM4) and the identities (LbCOM5) and (LbCOM6) yield the equalities −[n, [q, p]] = [n, [p, q]] and −[q, [n, p]] = [q, [p, n]]. These together with (LbCOM2)–(LbCOM5) allow to prove that ((σp 1, θp 1),(σp 2, θp 2)) does satisfy conditions (2.2.22)–(2.2.27) from Definition 2.2.11. Therefore, ψis well defined, while it is obviously K-linear. Concerning the preservation of the Leibniz bracket by ψ, due to (2.2.28) we know that [ψ(p), ψ(p0)] = ((σp 1σp0 1−σp0 1σp 1, θp 1σp0 1−σp0 1θp 1),(σp 2σp0 2−σp0 2σp 2, θp 2σp0 2−σp0 2θp 2)), and by definition ψ([p, p0]) = ((σ[p,p0] 1, θ[p,p0] 1),(σ[p,p0] 2, θ[p,p0] 2)). One can easily check that the corresponding components are equal by making use of (2.2.39), (2.2.40), (2.2.45) and (2.2.46). Hence, ψis a morphism of Leibniz algebras. Recall that ∆ϕ(m) = ((dmµ, Dmµ),(µdm, µDm)), ψη(m) = ((ση(m) 1, θη(m) 1),(ση(m) 2, θη(m) 2)), for any m∈m, but dmµ(n) = −ξ2(µ(n), m) = −[n, m] = −[n, η(m)] = ση(m) 1(n), Dmµ(n) = ξ1(m, µ(n)) = [m, n] = [η(m), n] = θη(m) 1(n), µdm(q) = −µξ2(q, m) = −[q, m] = −[q, η(m)] = ση(m) 2(q), µDm(q) = µξ1(m, q) = [m, q] = [η(m), q] = θη(m) 2(q), for all n∈n,q∈q, due to (LbM1a), (LbM1b), (LbM2a), (LbM2b) and the definition of the action of mon nand qvia η. Therefore, ∆ϕ=ψη. 82 2 Actors and modules over crossed modules It only remains to check the behaviour of (ϕ, ψ) regarding the action of pon m. Let m∈mand p∈p. Due to (2.2.29) and (2.2.30), [ψ(p), ϕ(m)] = (σp 1dm−dmσp 2, θp 1dm−dmθp 2), [ϕ(m), ψ(p)] = (dmσp 2−σp 1dm, Dmσp 2−σp 1Dm). On the other hand, by definition, we know that ϕ([p, m]) = (d[p,m], D[p,m]), ϕ([m, p]) = (d[m,p], D[m,p]). Directly from (LbM3a), (LbM3b), (LbM3c) and (LbM3d) one can easily confirm that the required identities between components hold. Let us write, as an example, the calculations in order to prove that the second component of [ϕ(m), ψ(p)] equals the second component of ϕ([m, p]). For any q∈q, (Dmσp 2−σp 1Dm)(q) = −Dm([q, p]) −σp 1(ξ1(m, q)) =−ξ1(m, [q, p]) + [ξ1(m, q), p], but according to (LbM3d), −ξ1(m, [q, p]) + [ξ1(m, q), p] = ξ1([m, p], q) = D[m,p](q). Hence, we can finally ensure that (ϕ, ψ) is a morphism of Leibniz crossed modules. Now let us show that it is necessary that at least one of the conditions (CON1)– (CON3) holds in order to prove the converse statement. Let us suppose that there is a morphism of crossed modules m p Bider(q,n) Bider(n,q, µ) η ϕψ ∆ (2.2.52) Given m∈mand p∈p, let us denote ϕ(m) by (dm, Dm) and ψ(p) by ((σp 1, θp 1),(σp 2, θp 2)), which satisfy conditions (2.2.16)–(2.2.18) from Definition 2.2.7 and conditions (2.2.20)– (2.2.27) from Definition 2.2.11 respectively. Also, due to the definition of ∆ (see Proposition 2.2.15), the commutativity of (2.2.52) can be expressed by the identity ((dmµ, Dmµ),(µdm, µDm)) = ((ση(m) 1, θη(m) 1),(ση(m) 2, θη(m) 2)) (2.2.53) for all m∈m. It is possible to define four bilinear maps, all of them denoted by [−,−], from p×nto n,n×pto n,p×qto qand q×pto q, given by [p, n] = θp 1(n),[n, p] = −σp 1(n), [p, q] = θp 2(q),[q, p] = −σp 2(q), 2.2.3 Actor crossed module of Leibniz algebras 83 for all n∈n,p∈p,q∈q. In order to prove that those maps define Leibniz actions of pon nand q, it is necessary to check that conditions (2.2.36)–(2.2.41) and (2.2.42)–(2.2.47) are satisfied by the corresponding maps. Conditions (2.2.36)–(2.2.38) (respectively (2.2.42)–(2.2.44)) follow easily from the fact that (σp 1, θp 1) (respectively (σp 2, θp 2)) is a biderivation of n(respectively q). Since ψis a Leibniz homomorphism, we know that, given p, p0∈p,ψ([p, p0]) = [ψ(p), ψ(p0)], that is ((σ[p,p0] 1, θ[p,p0] 1),(σ[p,p0] 2, θ[p,p0] 2)) = ((σp 1σp0 1−σp0 1σp 1, θp 1σp0 1−σp0 1θp 1), (σp 2σp0 2−σp0 2σp 2, θp 2σp0 2−σp0 2θp 2)). The identities between the first and the second (respectively the third and the fourth) components in those quadruples allow to confirm that (2.2.39) and (2.2.40) (respectively (2.2.45) and (2.2.46)) hold. As for conditions (2.2.41) and (2.2.47), it is fairly straightforward to check that [[p, p0], n]−[[p, n], p0] = θp 1σp0 1(n), [[p, p0], q]−[[p, q], p0] = θp 2σp0 2(q), while [p, [p0, n]] = θp 1θp0 1(n), [p, [p0, q]] = θp 2θp0 2(q), for all n∈n,p, p0∈p,q∈q. However, if at least one of the conditions (CON1)– (CON3) holds, due to Lemma 2.2.17 (i), θp 1σp0 1(n) = θp 1θp0 1(n) and θp 2σp0 2(q) = θp 2θp0 2(q). Therefore, we can ensure that there are Leibniz actions of pon both nand q, which induce actions of mon nand qvia η. The reader might have noticed that a fourth possible condition on (n,q, µ) could have been considered in order to guarantee the existence of the actions of pon nand qfrom the existence of the morphism of Leibniz crossed modules (ϕ, ψ). In fact, if [n,n] = nand Ann(q) = 0, the problem with conditions (2.2.41) and (2.2.47) could have been solved the same way. Nevertheless, this fourth condition does not guarantee that (ii) holds, as we will prove immediately below. Regarding (LbEQ1) and (LbEQ2), they follow directly from (2.2.21) (observe that, by hypothesis, ((σp 1, θp 1),(σp 2, θp 2)) is a biderivation of (n,q, µ) for any p∈p). Similarly, (LbCOM1)–(LbCOM6) follow almost immediately from (2.2.22)–(2.2.27). Hence, (i) holds. Concerning (ii), we can define ξ1(m, q) = Dm(q) and ξ2(q, m) = −dm(q) for any m∈m,q∈q. In this way, ξ1and ξ2are clearly bilinear. (LbM1a), (LbM1b), (LbM2a) and (LbM2b) follow immediately from the identity (2.2.53) and the fact that the actions of mon nand qare induced by the actions of pvia η. 84 2 Actors and modules over crossed modules Identities (LbM5a), (LbM5b) and (LbM5c) are a direct consequence of (2.2.16)– (2.2.18) (recall that, by hypothesis, (dm, Dm) is a biderivation from qto nfor any m∈m). Note that ϕis a Leibniz homomorphism, so ϕ([m, m0]) = [ϕ(m), ϕ(m0)] for m, m0∈ m, that is (d[m,m0], D[m,m0])=(dmµdm0−dm0µdm, Dmµdm0−dm0µDm). This identity, together with (LbM2a) and (LbM2b), allows to easily prove that (LbM4a) and (LbM4b) hold. Note that, since (ϕ, ψ) is a morphism of Leibniz crossed modules, ϕ([p, m]) = [ψ(p), ϕ(m)] and ϕ([m, p]) = [ϕ(m), ψ(p)] for all m∈m,p∈p. Due to the definition of the action of Bider(n,q, µ) on Bider(q,n) (see Theorem 2.2.16), we can write (d[p,m], D[p,m]) = (σp 1dm−dmσp 2, θp 1dm−dmθp 2), (d[m,p], D[m,p])=(dmσp 2−σp 1dm, Dmσp 2−σp 1Dm). (LbM3a), (LbM3b), (LbM3c) and (LbM3d) follow immediately from the previous identities. Regarding (LbM6a) and (LbM6b), directly from the definition of ξ1,ξ2and the actions of pon nand q, we have that ξ1(m, [p, q]) = Dmθp 2(q),[p, ξ1(m, q)] = θp 1Dm(q), −ξ1(m, [q, p]) = Dmσp 2(q),−[p, ξ2(q, m)] = θp 1dm(q), for all m∈m,p∈p,q∈q. Nevertheless, if at least one of the conditions (CON1)– (CON3) holds, due to Lemma 2.2.17 (ii), Dmθp 2(q) = Dmσp 2(q) and θp 1Dm(q) = θp 1dm(q). Hence, (ii) holds. Remark 2.2.19. A closer look at the proof of the previous theorem shows that neither conditions (LbM6a) and (LbM6b), nor the identities (2.2.41) and (2.2.47) (which correspond to the sixth axiom satisfied by the actions of pon nand qrespectively) are necessary in order to prove the existence of a morphism of crossed modules (ϕ, ψ) from (m,p, η)to Act(n,q, µ), under the hypothesis that (i) and (ii) hold. Actually, if we remove those conditions from (i) and (ii), the converse statement would be true for any Leibniz crossed module (n,q, µ), even if it does not satisfy any of the conditions (CON1)–(CON3). An early version of Theorem 2.2.18 did not include conditions (LbM6a) and (LbM6b), although (2.2.41) and (2.2.47) were considered from the very beginning, since it does not seem natural to ask for pto “almost” act on nand q. The problem is that (LbM6a),(LbM6b),(2.2.41) and (2.2.47) are essential in order to prove that (i) and (ii) as in Theorem 2.2.18 describe a set of derived actions of (m,p, η)on (n,q, µ), as we will show immediately below. This agrees with the idea of Act(n,q, µ)not being “good enough” to be the actor of (n,q, µ)in general, just as Bider(m)is not always the actor of a Leibniz algebra m. 2.2.3 Actor crossed module of Leibniz algebras 85 Example 2.2.20. Given a Leibniz crossed module (m,p, η), there is a morphism (ϕ, ψ): (m,p, η)→Act(m,p, η), with ϕ(m) = (dm, Dm)and ψ(p) = ((σp 1, θp 1),(σp 2, θp 2)), where dm(p) = −[p, m], Dm(p)=[m, p], and σp 1(m) = −[m, p], θp 1(m)=[p, m], σp 2(p0) = −[p0, p], θp 2(p0) = [p, p0], for all m∈m,p, p0∈p. Calculations in order to prove that (ϕ, ψ)is indeed a morphism of Leibniz crossed modules are fairly straightforward only by making use of the axioms satisfied by the actions of pon mtogether with the equivariance and the Peiffer identity. Of course, this morphism does not necessarily define a set of derived actions. Theorem 2.2.18, along with the result immediately bellow, shows that if (m,p, η)satisfies at least one of the conditions (CON1)–(CON3), then the previous morphism does define a set of derived actions of (m,p, η)on itself. Let (m,p, η) and (n,q, µ) be Leibniz crossed modules such that (i) and (ii) from Theorem 2.2.18 hold. Therefore, there are Leibniz actions of mon nand of pon q, so it makes sense to consider the semidirect products of Leibniz algebras nomand qop. Furthermore, we have the following result. Theorem 2.2.21. There is an action of the Leibniz algebra qopon the Leibniz algebra nom, given by [(q, p),(n, m)] = ([q, n]+[p, n] + ξ2(q, m),[p, m]),(2.2.54) [(n, m),(q, p)] = ([n, q]+[n, p] + ξ1(m, q),[m, p]),(2.2.55) for all (q, p)∈qop,(n, m)∈nom, with ξ1and ξ2as in Theorem 2.2.18. Moreover, the Leibniz homomorphism (µ, η): nom→qop, given by (µ, η)(n, m)=(µ(n), η(m)), for all (n, m)∈nomis a Leibniz crossed module together with the previous action. Proof. In order to prove that (2.2.54) and (2.2.55) define an action of qopon nomit is necessary to confirm that the following identities hold for any (n, m),(n0, m0)∈nom, 86 2 Actors and modules over crossed modules (q, p),(q0, p0)∈qop. [(q, p),[(n, m),(n0, m0)]] = [[(q, p),(n, m)],(n0, m0)] −[[(q, p),(n0, m0)],(n, m)], (2.2.56) [(n, m),[(q, p),(n0, m0)]] = [[(n, m),(q, p)],(n0, m0)] −[[(n, m),(n0, m0)],(q, p)], (2.2.57) [(n, m),[(n0, m0),(q, p)]] = [[(n, m),(n0, m0)],(q, p)] −[[(n, m),(q, p)],(n0, m0)], (2.2.58) [(n, m),[(q, p),(q0, p0)]] = [[(n, m),(q, p)],(q0, p0)] −[[(n, m),(q0, p0)],(q, p)],(2.2.59) [(q, p),[(n, m),(q0, p0)]] = [[(q, p),(n, m)],(q0, p0)] −[[(q, p),(q0, p0)],(n, m)],(2.2.60) [(q, p),[(q0, p0),(n, m)]] = [[(q, p),(q0, p0)],(n, m)] −[[(q, p),(n, m)],(q0, p0)].(2.2.61) Recall that the brackets in nomand qopare given respectively by [(n, m),(n0, m0)] = ([n, n0]+[m, n0]+[n, m0],[m, m0]) and [(q, p),(q0, p0)] = ([q, q0]+[p, q0]+[q, p0],[p, p0]) for all (n, m),(n0, m0)∈nom, (q, p),(q0, p0)∈qop. The procedure in order to check each of the identities is not complicated if one bears in mind the conditions satisfied by (m,p, η) and (n,q, µ) (see Theorem 2.2.18). Nevertheless, as an example, we show how to prove (2.2.58). Calculations for the rest of the identities are similar. Let (n, m),(n0, m0)∈nomand (q, p)∈qop. By routine calculations we get that [(n, m),[(n0, m0),(q, p)]] = ([n, [n0, q]] | {z } (1) +[n, [n0, p]] | {z } (2) +[n, ξ1(m0, q)] | {z } (3) +[m, [n0, q]] | {z } (4) +[m, [n0, p]] | {z } (5) +[m, ξ1(m0, q)] | {z } (6) +[n, [m0, p]] | {z } (7) ,[m, [m0, p]] | {z } (8) ), [[(n, m),(n0, m0)],(q, p)] = ([[n, n0], q] | {z } (10) +[[n, n0], p] | {z } (20) +[[n, m0], q] | {z } (30) +[[m, n0], q] | {z } (40) +[[m, n0], p] | {z } (50) +ξ1([m, m0], q) | {z } (60) +[[n, m0], p] | {z } (70) ,[[m, m0], p] | {z } (80) ), [[(n, m),(q, p)],(n0, m0)] = ([[n, q], n0] | {z } (100 ) +[[n, p], n0] | {z } (200 ) +[[n, q], m0] | {z } (300 ) +[ξ1(m, q), n0] | {z } (400 ) +[[m, p], n0] | {z } (500 ) +[ξ1(m, q), m0] | {z } (600 ) +[[n, p], m0] | {z } (700 ) ,[[m, p], m0] | {z } (800 ) ). Let us show that (i) = (i0)−(i00) for i= 1,...,8. It is immediate for i= 1,2,8 due to the action of qon nand the actions of pon nand m. For i= 5, the identity follows 2.2.3 Actor crossed module of Leibniz algebras 87 from the fact that the action mon nis defined via ηtogether with the equivariance of η. Namely, [m, [n0, p]] = [η(m),[n0, p]] = [[η(m), n0], p]−[[η(m), p], n0] = [[m, n0], p]−[η([m, p]), n0] = [[m, n0], p]−[[m, p], n0]. The procedure is similar for i= 7. For i= 3, it is necessary to make use of the Peiffer identity of µ, (LbM1b), the definition of the action of mon nand qvia ηand (LbCOM1): [n, ξ1(m0, q)] = [n, µξ1(m0, q)] = [n, [m0, q]] = [n, [η(m0), q]] = [[n, η(m0)], q]−[[n, q], η(m0)] = [[n, m0], q]−[[n, q], m0]. The conditions required in order to prove the identity for i= 4 are the same used for i= 3 except (LbCOM1), which is replace by (LbCOM2). Finally, for i= 6, due to (LbM4b) and the definition of the action of mon nvia η, we know that ξ1([m, m0], q) = [ξ1(m, q), m0]−[m, ξ2(q, m0)] = [ξ1(m, q), m0]−[η(m), ξ2(q, m0)], but applying (LbM6b), we get ξ1([m, m0], q) = [ξ1(m, q), m0]+[η(m), ξ1(m0, q)] = [ξ1(m, q), m0]+[m, ξ1(m0, q)], so (6) = (60)−(600) and (2.2.58) holds. Note that (LbM6a) and (LbM6b) are necessary in order to check (2.2.59) and (2.2.60) respectively. Checking that (µ, η) is indeed a Leibniz homomorphism follows directly from the definition of the action of mon nvia ηtogether with the conditions (LbEQ1) and (LbEQ2). Regarding the equivariance of (µ, η), given (n, m)∈nomand (q, p)∈qop, (µ, η)([(q, p),(n, m)]) = (µ, η)([q, n]+[p, n] + ξ2(q, m),[p, m]) = (µ([q, n]) + µ([p, n]) + µξ2(q, m), η([p, m])) = ([q, µ(n)] + [p, µ(n)] + [q, m],[p, η(m)]) = ([q, µ(n)] + [p, µ(n)] + [q, η(m)],[p, η(m)]) = [(q, p),(µ(n), η(m))], due to the equivariance of µand η, (LbEQ1), (LbM1a) and the definition of the action of mon qvia η. Similarly, but using (LbEQ2) and (LbM1b) instead of (LbEQ1) and (LbM1a), it can be proved that (µ, η)([(n, m),(q, p)]) = [(µ(n), η(m)),(q, p)]. The Peiffer identity of (µ, η) follows easily from the homonymous property of µ and η, the definition of the action of mon nvia ηand the conditions (LbM2a) and (LbM2b). Definition 2.2.22. The Leibniz crossed module (nom,qop,(µ, η)) is called the semidirect product of the Leibniz crossed modules (n,q, µ)and (m,p, η). 88 2 Actors and modules over crossed modules Note that the semidirect product determines an obvious split extension of (m,p, η) by (n,q, µ) (0,0,0) (n,q, µ) (nom,qop,(µ, η)) (m,p, η) (0,0,0) Conversely, any split extension of (m,p, η) by (n,q, µ) is isomorphic to their semidirect product, where the action of (m,p, η) on (n,q, µ) is induced by the splitting morphism. Now we are in a position to write the following definition. Definition 2.2.23. If (m,p, η)and (n,q, µ)are Leibniz crossed modules and at least one of the following conditions holds, 1. Ann(n) = 0 = Ann(q), 2. Ann(n) = 0 and [q,q] = q, 3. [n,n] = nand [q,q] = q, an action of the crossed module (m,p, η)on (n,q, µ)is a morphism of Leibniz crossed modules from (m,p, η)to Act(n,q, µ). In other words, under one of those conditions, Act(n,q, µ)is the actor of (n,q, µ). Example 2.2.24. (i) Given a Leibniz algebra q, it can be regarded as a Leibniz crossed module in two obvious ways, ({0},q,0) and (q,q,idq). It is easy to check that Act({0},q,0) ∼ = ({0},Bider(q),0) and Act(q,q,idq)∼ =(Bider(q),Bider(q),id). (ii) Every Lie crossed module (n,q, µ)can be regarded as a Leibniz crossed module (see Subsection 3.4.2). Note that in this situation, both the multiplication and the action are antisymmetric. Therefore, given (d, D)∈Bider(q,n), both dand Dare elements in Der(q,n). Additionally, if we assume that at least one of the conditions from the previous lemma holds, then either Ann(n) = 0 or [q,q] = q. In this situation, one can easily derive from (2.2.18) that Bider(q,n) = {(d, d)|d∈Der(q,n)}. Besides, the bracket in Bider(q,n)becomes antisymmetric and, as a Lie algebra, it is isomorphic to Der(q,n). Similarly, Bider(n,q, µ)is a Lie algebra isomorphic to Der(n,q, µ)and Act(n,q, µ)is a Lie crossed module isomorphic to Act(n,q, µ). 2.2.4 On the actor crossed module of associative algebras and dialgebras A closer look at the procedure followed in the previous subsection in order to construct a general actor for a crossed module of Leibniz algebras, makes us wonder if a similar approach could lead us to the construction of 2-dimensional analogues to the associative algebra of bimultipliers and the associative dialgebra of tetramultipliers. It does not seem reckless to think that, given a crossed module of algebras (respectively dialgebras) (B, A, ρ) (respectively (L, D, µ)), (Bim(A, B),Bim(B, A, ρ),∆) (respectively 2.3 Modules over crossed modules 89 (Tetra(D, L),Tetra(L, D, µ),∆0)) could be a good candidate for general actor, or even actor under certain conditions. Of course it would be necessary to give a proper definition of all the objects involved. Furthermore, the amount of equations involved in the definition of an action of a crossed module of dialgebras would probably be enormous, around five times what we have for Leibniz algebras. 2.3 Modules over crossed modules It is a classical fact that the categories of (left or right) modules over a Lie algebra and over its universal enveloping algebra are equivalent. Since we intend to establish the analogous equivalence for the categories of modules over a Lie crossed module and over its universal enveloping crossed module, notion that will be explored in the last chapter, we need a proper definition of left modules over a crossed module of Lie and associative algebras. Recall that Beck [7] introduced a convenient notion of coefficient module to be used in (co)homology theories. That concept makes sense in a broad context and recovers the usual notions of modules in familiar settings: for groups, commutative algebras and Lie algebras, these are left modules; for associative algebras, the appropriate notion is that of bimodule. By definition, given an object Cof a category C, a Beck module over Cis an abelian group object in the slice category C/C. If Cis a category of interest in the sense of Orzech [74], then Beck modules are equivalent to split extensions with singular kernel [74, Theorem 2.7]. This is the case for all the aforementioned familiar categories. Moreover, the description of crossed modules of groups as cat1-groups makes XGr into a category of interest (see for instance [75]). Nevertheless, the same assertion fails in the case of Lie crossed modules. Concretely, the category XLie satisfies all the axioms of a category of interest except one (axiom (d) from Definition 1.1.1), which is replaced by a new axiom (see the details in [21] for precrossed modules of Lie algebras). However, since that condition is not used in the proof of [74, Theorem 2.7], we can still apply this general result to XLie and identify a module over a Lie crossed module (m,p, ν) with a split extension in XLie, (0,0,0) (n,q, µ) (m0,p0, ν0) (m,p, ν) (0,0,0) where the kernel (n,q, µ) is an abelian crossed module of Lie algebras, that is n,qare abelian Lie algebras and the action of qon nis trivial. Bearing in mind the discussion at the end of Subsection 2.2.2, it makes sense to consider the following definition. Definition 2.3.1. Let (m,p, ν)be a Lie crossed module. A left (m,p, ν)-module is an abelian Lie crossed module (n,q, µ)together with a morphism of crossed modules (ϕ, ψ)from (m,p, ν)to Act(n,q, µ). 96 3 Adjunctions between categories of crossed modules 3.1 XGr vs XAs1 Given a unital algebra Aits subset of all invertible elements Ug(A) forms a group called the group of units of the algebra A. Besides, given a morphism of unital algebras f:A→B, it is possible to consider the group homomorphism Ug(f): Ug(A)→Ug(B), where Ug(f) = f|Ug(A). These assignments define a functor Ug:As1→Gr called the unit group functor. The left adjoint to Ugis the functor K:Gr →As1, which sends every group Gto its group algebra K(G), that is the free module over Kon the underlying set of G with the multiplication defined on the basis elements by the group operation in G. Given a group homomorphism f:G→H, its corresponding morphism of algebras is K(f): K(G)→K(H), which is defined by extending fto the elements of K(G) by linearity. In this section we recall the construction of the natural generalization of these functors, XGr XAs1 XK XUg which can be found in [22]. 3.1.1 The crossed module of units Let us begin with the definition of XUg. Given a crossed module of unital algebras (B, A, ρ), consider its corresponding cat1-algebra as described in the proof of Proposition 1.2.34: BoA A σ τ with σ(b, a) = aand τ(b, a) = ρ(b) + afor all (b, a)∈BoA. Note that the unit of BoAis (0,1). Lemma 3.1.1. Given an object (B, A, ρ)in XAs1and its corresponding cat1-algebra (BoA, A, σ, τ), (i) Every element in Ker Ug(σ)is of the form (b, 1), with b∈B, and there is b0∈B such that bb0+b0+b=0=b0b+b+b0.(3.1.1) (ii) (Ug(BoA),Ug(A),Ug(σ),Ug(τ)) is a cat1-group. Proof. (i) Given (b, a)∈Ker Ug(σ), 1 = Ug(σ)(b, a) = a. Therefore (b, a)=(b, 1). On the other hand, (b, 1) ∈Ug(BoA), so there is (b0, a0)∈BoAsuch that (b, 1)(b0, a0) = (0,1) = (b0, a0)(b, 1). Directly from the definition of the multiplication in BoA, we get that a0= 1 and bb0+b0+b=0=b0b+b+b0. (ii) The identities Ug(σ)|Ug(A)=Ug(τ)|Ug(A)= idUg(A)follow immediately from σ|A=τ|A= idAand the definition of the functor Ug. 3.1.1 The crossed module of units 97 It only remains to prove that [Ker Ug(σ),Ker Ug(τ)] = 1. Let (b1,1) ∈Ker Ug(σ) and (b2, a2)∈Ker Ug(τ). Note that 1 = Ug(τ)(b2, a2) = ρ(b2) + a2, so a2= 1 −ρ(b2). Using the definition of the product in BoAand (XAs2), we get that: (b1,1)(b2,1−ρ(b2)) = (b1b2+b2+b1−b1ρ(b2),1−ρ(b2)) = (b2+b1,1−ρ(b2)) = (b2b1+b1−ρ(b2)b1+b2,1−ρ(b2)) = (b2,1−ρ(b2))(b1,1). Therefore [Ker Ug(σ),Ker Ug(τ)] = 1. Now we can define XUg(B, A, ρ) = (Ker Ug(σ),Ug(A),Ug(τ)|Ker Ug(σ)) for any crossed module of unital algebras (B, A, ρ), where (Ug(BoA),Ug(A),Ug(σ),Ug(τ)) is the result of applying Ugto the cat1-algebra (BoA, A, σ, τ). Sometimes we will write Ug(τ) instead of Ug(τ)|Ker Ug(σ)for ease of notation. Remark 3.1.2. Note that Ug(A)can be regarded as a subgroup of Ug(B×A)via the inclusion a7→ (0, a)and the action of Ug(A)on Ker Ug(σ)is defined in the proof of Proposition 1.2.12 as conjugation. Namely, given a∈Ug(A)and (b, 1) ∈Ker Ug(σ), a(b, 1) = (0,a)(b, 1) = (0, a)(b, 1)(0, a−1) = (ab, a)(0, a−1) = (aba−1,1). For any morphism of crossed modules of algebras (ϕ, ψ): (B, A, ρ)→(B0, A0, ρ0), the corresponding morphism of crossed modules of groups XUg(ϕ, ψ) is given by: Ker Ug(σ)Ug(A) Ker Ug(σ0)Ug(A0). Ug(τ) (ϕ,1) ψ Ug(τ0) Recall that a group G(resp. a unital algebra A) can be regarded as a crossed module of groups (resp. a crossed module of unital algebras) in two obvious ways: via the trivial map 1: {1} → G(resp. 0: {0} → A) or via the identity map idG:G→G (resp. idA:A→A) with the action of G(resp. of A) on itself defined by conjugation (resp. by multiplication). See Example 1.2.5 (i) for groups and Example 1.2.28 (i) for algebras. We have the functors: E0,E1:Gr XGr E0 0,E0 1:As1XAs1 where E0(G) = ({1}, G, 1), E1(G) = (G, G, idG), E0 0(A) = ({0}, A, 0) and E0 1(A) = (A, A, idA), given any group Gand any unital algebra A. 98 3 Adjunctions between categories of crossed modules It is easy to check that the following diagram is commutative. As1XAs1 Gr XGr . E0 0 UgXUg E0 Note that Ker Ug(σ) = {1}for Ug(σ): Ug({0}oA)→Ug(A), with Ug(σ)(0, a) = a, for all a∈A. Concerning the embeddings E1and E0 1, we have the following result: Proposition 3.1.3 ([22]).There is a natural isomorphism of functors XUg◦E0 1∼ =E1◦Ug. Proof. Let A∈As1. We need to prove that XUg(A, A, idA) is naturally isomorphic to (Ug(A),Ug(A),idUg(A)). According to the foregoing definition of XUg, we first need to consider the cat1-algebra (AoA, A, σ, τ), with σ(a, a0) = a0and τ(a, a0) = a+a0 for all (a, a0)∈AoA. Then we apply Ugto (AoA, A, σ, τ) and XUg(A, A, idA) = (Ker Ug(σ),Ug(A),Ug(τ)). It is clear that (Ug(τ),idUg(A)) is a morphism of crossed modules of groups from XUg(A, A, idA) to (Ug(A),Ug(A),idUg(A)) (see Example 1.2.8 (iii)). Note that every element in Ker Ug(σ) is of the form (a, 1), with a∈A(see Lemma 3.1.1 (i)) and Ug(τ)(a, 1) = a+1. It is easy to check that φ:A→AoA,a7→ (a−1,1) is a morphism of algebras. Therefore Ug(φ) = φ|Ug(A)is a group homomorphism from Ug(A) to Ug(AoA). Furthermore, φ(Ug(A)) ⊂Ker Ug(σ). The diagram Ug(A)Ug(A) Ker Ug(σ)Ug(A) idUg(A) φidUg(A) Ug(τ) is clearly commutative. Additionally, given a1, a2∈Ug(A), φ(a1a2) = φ(a1a2a−1 1)=(a1a2a−1 1−1,1) = (a1a2a−1 1−a1a−1 1,1) = (a1(a2−1)a−1 1,1) = a1(a2−1,1) = a1φ(a2), so (φ, idUg(A)) is a morphism of crossed modules of groups. It is easy to check that (φ, idUg(A)) is the inverse of (Ug(τ),idUg(A)) and the naturality of (Ug(τ),idUg(A)) is obvious. 3.1.2 The group algebra crossed module 99 3.1.2 The group algebra crossed module Let us now recall the construction of the left adjoint to XK. Let (H, G, ∂) be a crossed module of groups and consider its corresponding cat1-group as in the proof of Proposition 1.2.12, that is HoG G s t with s(h, g) = gand t(h, g) = ∂(h)gfor all (h, g)∈HoG. If we apply Kto the previous diagram we get K(HoG)K(G) K(s) K(t) Although it is true that K(s)|K(G)=K(t)|K(G)= idK(G), in general, the second condition for cat1-algebras (CAs2) is not satisfied. For instance, let us take h∈H\ {1}and g∈Gsuch that gh6=h−1. It is clear that v= (h, g)−(1, g)∈Ker K(s) and w= (h, g)−(1, ∂(h)g)∈Ker K(t). Besides, vw = (hgh, gg)−(gh, gg)−(h, g∂(h)g) + (1, g∂(h)g), is a linear combination of elements from the basis of K(HoG) with non-zero coefficients due to the condition gh6=h−1. Hence, Ker K(s) Ker K(t)6= 0 in general. Nevertheless, we can consider the quotient K(HoG) = K(HoG)/X, where X= Ker K(s) Ker K(t) + Ker K(t) Ker K(s), together with the induced morphisms K(s) and K(t). In this way, the diagram K(HoG)K(G) K(s) K(t) is clearly a cat1-algebra. Note that X⊂Ker K(t) and K(t)|K(G)= idK(G), since t|G= idG. Given v, w ∈K({1}oG)≃K(G), if v−w∈X, then 0 = K(t)(v−w) = v−w. Therefore K(G) can be regarded as a subalgebra of K(HoG). We can now define XK(H, G, ∂) as the crossed module of algebras corresponding to the previous cat1-algebra, that is (Ker K(s),K(G),K(t)|Ker K(s)). Sometimes we will write K(t) instead of K(t)|Ker K(s)to ease notation. For any morphism of crossed modules (ϕ, ψ): (H, G, ∂)→(H0, G0, ∂0), XK(ϕ, ψ) is given by Ker K(s)K(G) Ker K(s0)K(G0) K(t) K(ϕ,ψ)|Ker K(s)K(ψ) K(t0) 100 3 Adjunctions between categories of crossed modules where K(ϕ, ψ) is the algebra homomorphism induced by K(ϕ, ψ), which is itself the K-linear extension of (ϕ, ψ): HoG→H0oG0, given by (ϕ, ψ)(h, g) = (ϕ(h), ψ(g)) for all (h, g)∈HoG. The functor XK :XGr →XAs1is a natural generalization of the functor K, in the sense that the following diagram commutes, Gr XGr As1XAs1. E0 K XK E0 0 In fact, given Ga group, it is clear that E0(K(G)) = ({0},K(G),0). Now, let us follow the steps in the construction of XK({1}, G, 1). First we consider the cat1-group {1}oG G s t with s(1, g) = t(1, g) = g. Then we apply the functor K, K({1}oG)K(G). K(s) K(t) It is clear that K(s) = K(t) is an isomorphism between the algebras K({1}oG) and K(G). Hence Ker K(s) = Ker K(t) = {0}and (K({1}oG),K(G),K(s),K(t)) is a cat1algebra. Moreover XK({1}, G, 1) = (Ker K(t),K(G),K(t)|Ker K(t)) = ({0},K(G),0). Nevertheless, for the functor XK there is no analogue to Proposition 3.1.3, that is we claim that XK ◦E1E0 1◦K. Let us first illustrate it with an example. Let G={e, x}be the group with two elements and K=R.E0 1(K(G)) = (K(G),K(G),idK(G)), where K(G) is a vector space over Rof dimension 2, endowed with the product resulting from extending by bilinearity the group operation in G. A base of K(G) is {e, x}. We will denote e= (1,0) and x= (0,1). Let us now construct XK(G, G, idG). First, we need to consider the cat1-group GoG G, s t with s(e, e) = e, t(e, e) = e, s(e, x) = x, t(e, x) = x, s(x, e) = e, t(x, e) = x, s(x, x) = x, t(x, x) = e. 3.1.2 The group algebra crossed module 101 The next step is to apply Kon that diagram. In this way, we get: K(GoG)K(G), K(s) K(t) with K(s) and K(t) as a result of extending by linearity sand trespectively. A basis for K(GoG) is given by {(e, e),(e, x),(x, e),(x, x)}. We will denote (e, e) = (1,0,0,0), (e, x) = (0,1,0,0), (x, e) = (0,0,1,0), (x, x) = (0,0,0,1). Bearing in mind the previous notation, K(s)(1,0,0,0) = (1,0),K(t)(1,0,0,0) = (1,0), K(s)(0,1,0,0) = (0,1),K(t)(0,1,0,0) = (0,1), K(s)(0,0,1,0) = (1,0),K(t)(0,0,1,0) = (0,1), K(s)(0,0,0,1) = (0,1),K(t)(0,0,0,1) = (1,0). Hence, K(s) = 1010 0101and K(t) = 1001 0110. Now we can easily calculate Ker K(s) and Ker K(t): Ker K(s) = {(a1, a2, a3, a4)|a1=−a3, a2=−a4, ai∈R,for i= 1,2,3,4} ={(a1, a2,−a1,−a2)|a1, a2∈R} =h(1,0,−1,0),(0,1,0,−1)i, Ker K(t) = {(a1, a2, a3, a4)|a1=−a4, a2=−a3, ai∈R,for i= 1,2,3,4} ={(a1, a2,−a2,−a1)|a1, a2∈R} =h(1,0,0,−1),(0,1,−1,0)i. Note that Forrester-Barker gives in [45] an explicit description of one basis for Ker K(s) and for Ker K(t). Those bases agree with the ones described above. Namely, according 102 3 Adjunctions between categories of crossed modules to Forrester-Barker, Ker K(s) = h{(g, g0)−(e, g0)|g, g0∈G, g 6=e}i =h{(x, e)−(e, e),(x, x)−(e, x)}i =h(0,0,1,0) −(1,0,0,0),(0,0,0,1) −(0,1,0,0)i =h(−1,0,1,0),(0,−1,0,1)i, Ker K(t) = h{(g, g0)−(e, gg0)|g, g0∈G, g 6=e}i =h{(x, e)−(e, x),(x, x)−(e, e)}i =h(0,0,1,0) −(0,1,0,0),(0,0,0,1) −(1,0,0,0)i =h(0,−1,1,0),(−1,0,0,1)i. Note that Ker K(s)+Ker K(t) has dimension 3 and Ker K(s)∩Ker K(t) = h(1,1,−1,−1)i. Observe that the product in K(GoG) and the product in K(G) are both commutative, since Gis an abelian group. Therefore X= Ker K(s) Ker K(t)+Ker K(t) Ker K(s) = Ker K(s) Ker K(t). Fairly straightforward calculations show that Ker K(s) Ker K(t) = h(1,1,−1−1)iwhich agrees with what we would get if we followed the description given by Forrester-Barker. In order to get a cat1-algebra, we need to consider the diagram K(GoG)/X K(G), K(s) K(t) with K(s) and K(t) induced by K(s) and K(t). Note that K(GoG)/X has dimension 3. We can consider the basis B={(0,1,0,0) + X, (0,0,1,0) + X, (0,0,0,1) + X}. Let us give an explicit basis for Ker K(s): Ker K(s) = {v+X∈K(GoG)/X |K(s)(v+X) = (0,0)}, but X= Ker K(s)∩Ker K(t)⊂Ker K(s), so Ker K(s) = {v+X∈K(GoG)/X |K(s)(v) = (0,0)}= Ker K(s)/X. Moreover, (−1,0,1,0)+X= (0, a, b, c)+X⇔(−1,−a, 1−b, −c)∈X⇔a= 1, b = 0, c =−1, (0,−1,0,1) + X= (0, a, b, c) + X⇔(0,−1−a, −b, 1−c)∈X ⇔a=−1, b = 0, c = 1. Hence, Ker K(s) = h(1,0,−1)Biand it has dimension 1. In the following diagram, 3.1.2 The group algebra crossed module 103 Ker K(s)K(G) K(G)K(G) K(t) K(t)idK(G) idK(G) Ker K(s) is a R-vector space of dimension 1, while K(G) has dimension 2. Therefore XK(G, G, idG)(K(G),K(G),idK(G)). The problem shown in the previous example persists in a general situation. Let us now assume that Gis any group, not necessarily the one with to elements. The first step in order to construct XK(G, G, idG) is to take the cat1-group GoG G, s t with s(g, g0) = g0and t(g, g0) = gg0. Consider the morphism of groups :G→GoG, (g) = (g, 1). It is clear that s((g)) = 1 and t((g)) = gfor every g∈G. Now, if we use the functor K, we get: K(G)K(GoG)K(G). K()K(s) K(t) It is clear that K(t)K() = idK(G), while K(s)(K()(Pλg)) = K(s)(Pλg) = K(1)(Pλg) =Pλ1 for any formal linear combination of elements in G. Note that K(s)K() is not the trivial map unless K={0}. Now, if we consider X= Ker K(s) Ker K(t) + Ker K(s) Ker K(t), the diagram K(GoG)/X K(G), K(s) K(t) is a cat1-algebra, with K(s) and K(t) induced by K(s) and K(t) respectively. The crossed module of algebras XK(G, G, idG) is defined as (Ker K(s),K(G),K(t)|Ker K(s)). As we stated previously, in Forrester-Barker’s PhD thesis [45] there is and explicit description of a basis for Ker K(s)⊂K(GoG), namely {(g, g0)−(1, g0)|g0∈G, g ∈ G\ {1}}. He also proves that {(g, g0)−(1, g0) + X|g∈G\ {1}, g0∈G}is a set of generators of Ker K(s). Let (g, g0)−(1, g0)+X∈Ker K(s). Then K(t)((g, g0)−(1, g0)+X) = gg0−g0. Now, if we apply πK() to gg0−g0, with πthe projection π:K(GoG)→K(GoG)/X, we get πK()(gg0−g0)=(gg0,1) −(g0,1) + X, but (gg0,1) −(g0,1) ∼(g, g0)−(1, g0). Actually (gg0,1) −(g0,1) −(g, g0) + (1, g0) = ((g, 1) −(1,1))((g0,1) −(1, g0)), with (g, 1) −(1,1) ∈Ker K(s) and (g0,1) −(1, g0)∈Ker K(t). Hence (gg0,1) −(g0,1) − (g, g0) + (1, g0)∈X= Ker K(s) Ker K(t) + Ker K(t) Ker K(s). 104 3 Adjunctions between categories of crossed modules The previous calculations show that πK()K(t)|Ker K(s)= idKer K(s). Observe that for K(t)|Ker K(s)to be an isomorphism between Ker K(s) and K(G), it should be a surjective morphism. Therefore πK() should take values in Ker K(s) for every element in K(G). In that case, 0 = K(s)πK() = K(s)K(), which is not true unless K={0}. Therefore XK ◦E1E0 1◦K. 3.1.3 Adjunction between XGr and XAs1 The following result is a natural generalization of the well-known classical adjunction between the categories Gr and As1. Theorem 3.1.4 ([22]).The functor XK is left adjoint to the functor XUg. Proof. Given (H, G, ∂) in XGr and (B, A, ρ) in XAs1, we have to construct a natural bijection HomXGr (H, G, ∂),XUg(B, A, ρ)∼ =HomXAs1XK(H, G, ∂),(B, A, ρ). Let (ϕ, ψ)∈HomXGr (H, G, ∂),XUg(B, A, ρ), that is H G Ker Ug(σ)Ug(A) ∂ ϕψ Ug(τ) such that Ug(τ)ϕ=ψ∂ and ϕ(gh) = ψ(g)ϕ(h) for all h∈H,g∈G. We can consider the corresponding morphism of cat1-groups by using the functor catGr as defined in the proof of Proposition 1.2.12: HoG G Ker Ug(σ)oUg(A)Ug(A) Ug(BoA)Ug(A). s t (ϕ,ψ) ϕ0 ψ ˜s ˜ t ≃idUg(A) Ug(σ) Ug(τ) Note that the isomorphism in the diagram above is explicitly described at the end of the proof of Proposition 1.2.12, as well as ˜sand ˜ t. 3.1.3 Adjunction between XGr and XAs1105 Since the functor Kis left adjoint to the functor Ug, we have the induced commutative diagrams of algebras K(HoG)K(G) BoA A K(s) ϕ0∗ ψ∗ σ and K(HoG)K(G) BoA A. K(t) ϕ0∗ ψ∗ τ Due to the identity Ker σKer τ+ Ker τKer σ= 0, we have a uniquely defined morphism of cat1-algebras K(HoG)/X K(G) BoA A K(s) K(t) ϕ0∗ ψ∗ σ τ where X= Ker K(s) Ker K(t) + Ker K(t) Ker K(s). Finally, we can make use the functor XmAs, which is described in the proof of Proposition 1.2.34, in order to get a uniquely defined homomorphism (ϕ∗∗, ψ∗) in HomXAs1XK(H, G, ∂),(B, A, ρ): Ker K(s)K(G) Ker σ A B A. K(t) ϕ0∗ ϕ∗∗ ψ∗ τ ≃idA ρ Note that in the diagram above ϕ0∗ =ϕ0∗|Ker K(s), Ker σ=Bo{0}and τ=τ|Ker σ=ρ. Now, let (φ, χ)∈HomXAs1XK(H, G, ∂),(B, A, ρ), that is Ker K(s)K(G) B A K(t) φχ ρ such that ρφ =χK(t) and φpreserves the action of K(G) on Ker K(s) via χ. We can consider the corresponding morphism of cat1-algebras by using the functor 112 3 Adjunctions between categories of crossed modules (U(t)|Ker U(s),idU(p)), which is given by U(p)U(p) Ker U(s)U(p). idU(p) πU()idU(p) U(t) Note that we have already proved that the above diagram commutes. Besides, given p, p0∈p, if we consider them as elements in U(p), πU()(p⊗p0)=(p, 0)⊗(p0,0)+X= (0, p)⊗(p0,0) + X= (p, 0) ⊗(0, p0) + X, so πU() clearly preserves the action of U(p) on itself via idU(p)when extended to all the elements in U(p). Finally, (U(t)|Ker U(s),idU(p)) is natural, as shown in the diagram below, where α:p→p0is a Lie homomorphism. Note that U(α, α) is actually U(α, α)|Ker U(s). Ker U(s)U(p) Ker U(s0)U(p0) U(p)U(p) U(p0)U(p0) U(t) U(t) U(α,α) idU(p) U(α) U(t0) U(t0)idU(p0) idU(p) U(α) U(α) idU(p0) 3.2.3 Adjunction between XLie and XAs The following result is a natural generalization of the well-known classical adjunction between the categories Lie and As. Theorem 3.2.5. The functor XU is left adjoint to the Liezation functor XLieAs. Proof. Given (m,p, ν) in XLie and (B, A, ρ) in XAs, we have to construct a natural bijection HomXLie (m,p, ν),XLieAs(B, A, ρ)∼ =HomXAs XU(m,p, ν),(B, A, ρ). 3.2.3 Adjunction between XLie and XAs 113 Let (ϕ, ψ)∈HomXLie (m,p, ν),XLieAs(B, A, ρ), that is m p LieAs(B)LieAs(A) ν ϕψ ρ such that ρϕ =ψν and ϕpreserves the action of pon mvia ψ. We can consider the following morphism of cat1-Lie algebras by using the functor catLie as defined in the proof of Proposition 1.2.23, as well as Lemma 3.2.3: mop p LieAs(BoA)LieAs(A) s t ϕ0ψ σ τ where ϕ0(m, p) = (ϕ(m), ψ(p)) for all (m, p)∈mop. Since the functor Uis left adjoint to the functor LieAs, we have the induced commutative diagrams of algebras U(mop)U(p) BoA A U(s) ϕ0∗ ψ∗ σ and U(mop)U(p) BoA A. U(t) ϕ0∗ ψ∗ τ Besides, we have a uniquely defined morphism of cat1-algebras: U(mop)/X U(p) BoA A U(s) U(t) ϕ0∗ ψ∗ σ τ where X= Ker U(s) Ker U(t) + Ker U(t) Ker U(s). Finally, we can use the functor XmAs, described in the proof of Proposition 1.2.34, to get a uniquely defined morphism (ϕ∗∗, ψ∗) in HomXAs XU(m,p, ν),(B, A, ρ): Ker U(s)U(p) Ker σ A B A. U(t) ϕ0∗ ϕ∗∗ ψ∗ τ ≃idA ρ 114 3 Adjunctions between categories of crossed modules Note that in the diagram above ϕ0∗ =ϕ0∗|Ker U(s), Ker σ=Bo{0}and τ=τ|Ker σ=ρ. Now, let (φ, χ)∈HomXAs XU(m,p, ν),(B, A, ρ), that is Ker U(s)U(p) B A U(t) φχ ρ such that ρφ =χU(t) and φpreserves the action of U(p) on Ker U(s) via χ. We can consider the corresponding morphism of cat1-algebras by using the functor catAs as defined in the proof of Proposition 1.2.34. U(mop)/X U(p) Ker U(s)oU(p)U(p) BoA A. U(s) U(t) ≃ φ0 idU(p) ˜σ ˜τ (φ,χ)χ σ τ Observe that the isomorphism in the diagram above is explicitly described at the end of the proof of Proposition 1.2.34, as well as ˜σand ˜τ. Let φ=φ0π, with π:U(mop)→U(mop)/X the canonical projection. Hence, we have the commutative diagrams of algebras U(mop)U(p) BoA A U(s) φχ σ and U(mop)U(p) BoA A. U(t) φχ τ Since the functor Uis left adjoint to the functor LieAs, we have the morphism of cat1-Lie algebras mop p LieAs(BoA)LieAs(A). s t φ∗χ∗ σ τ Finally, we can use the functor XmLie, described in the proof of Proposition 1.2.23, 3.2.3 Adjunction between XLie and XAs 115 to get a uniquely defined morphism (φ∗∗, χ∗) in HomXLie (m,p, ν),XLieAs(B, A, ρ): m p Ker sp Ker σLieAs(A) LieAs(B)LieAs(A). ν ≃ φ∗∗ idp t φ∗|Ker sχ∗ τ ≃idLieAs(A) ρ Note that in the diagram above Ker s=mo{0},t=t|Ker s=ν, Ker σ=LieAs(B)o {0}and τ=τ|Ker σ=ρ. Bearing in mind the previous theorem along with Propositions 1.2.20 and 1.2.31, one can easily deduce the following result. Theorem 3.2.6. The inner and outer squares in the following diagrams are commutative or commute up to isomorphism for i= 0,1. As Lie As Lie XAs XLie XAs XLie ⊥ LieAs I0 i a U Ii` ⊥ LieAs I0 ia U Ii ` > XLieAs Φ0 i XU Φi > XLieAs Φ0 i+1 XU Φi+1 (3.2.3) Proof. Let us begin with the first diagram. Directly from the definition of the functors involved, XLieAs ◦I0 i=Ii◦LieAs for i=0,1. Besides, from the adjunctions described in the first diagram, we get that U◦Φiis left adjoint to Ii◦LieAs, while Φ0 i◦XU is left adjoint to XLieAs ◦I0 i, for i= 0,1. Hence, U◦Φi∼ =Φ0 i◦XU for i= 0,1. Regarding the second diagram, the commutativity of the outer square is obvious for i= 0, while in Proposition 3.2.4 we proved that there is a natural isomorphism XU ◦I1∼ =I0 1◦U. Therefore, by a similar reasoning to the one used for the first diagram, we get that LieAs ◦Φ0 i+1 ∼ =Φi+1 ◦XLieAs for i= 0,1. Recall that in Section 2.3 we gave an explicit definition of left modules over crossed modules of Lie and associative algebras. 116 3 Adjunctions between categories of crossed modules Lemma 3.2.7. Let δ:V→Wbe a K-module homomorphism, regarded as an abelian crossed module of algebras (Lie algebras). Then, using the same notations as in Lemma 2.3.4, the Lie crossed module XLieAs HomK(W, V ),End(V, W, δ),Γcoincides with the actor crossed module Act(V, W, δ) = Der(W, V ),Der(V, W, δ),∆. Proof. Since Vand Ware considered as abelian Lie algebras together with the trivial action of Won V, it is clear that Der(W, V ) = Lie HomK(W, V ), Der(V, W, δ) = LieAs End(V, W, δ)and ∆ = LieAs(Γ). Moreover, the Lie action of Der(V, W, δ) on Der(W, V ) is induced by the algebra action of End(V, W, δ) on HomK(W, V ). Theorem 3.2.8. Let (m,p, ν)be a Lie crossed module. Then the categories of left (m,p, ν)-modules and left XU(m,p, ν)-modules are isomorphic. Proof. By using Theorem 3.2.5 and Lemma 3.2.7, left (m,p, ν)-module structures on aK-module homomorphism δ:V→Ware in bijective correspondence with left XU(m,p, ν)-module structures on it: HomXLie (m,p, ν),Act(V, W, δ) = HomXLie (m,p, ν),XLieAs HomK(W, V ),End(V, W, δ),Γ ≈HomXAs XU(m,p, ν),HomK(W, V ),End(V, W, δ),Γ. Due to the conditions satisfied by a morphism between modules over a crossed module of Lie and associative algebras (see (2.3.1)–(2.3.3) and (2.3.9)–(2.3.11)), it is easy to check that this correspondence is functorial. Finally, let us remark that right modules (over crossed modules of Lie and associative algebras) could be defined similarly and could equally be used everywhere instead of left modules. 3.3 XLb vs XDias As explained by Loday in [65], a Leibniz algebra is a non-commutative version of a Lie algebra. When we replace Lie algebras by Leibniz algebras, the role of associative algebras is played by associative dialgebras. Any dialgebra Dbecomes a Leibniz algebra with the bracket given by [x, y] = xa y−y`xfor all x, y ∈D. Moreover, any dialgebra homomorphism f:D→Lis a Leibniz algebra homomorphism if we consider the bracket previously defined both in Dand L. Thus we have a functor Lb :Dias →Lb. The functor Lb admits a left adjoint, Ud:Lb →Dias, which assigns to a Leibniz algebra pits universal enveloping dialgebra Ud(p) (see [65]), which is defined as the quotient T(p)⊗p⊗T(p)/J of the free dialgebra over the underlying K-module of p 3.3.1 From XDias to XLb 117 by the ideal Jgenerated by p1ap2−p2`p1−[p1, p2], for all p1, p2∈p. Note that the two products in T(p)⊗p⊗T(p) are induced by (p−n· · · p−1⊗p0⊗p1· · · pm)a(q−s· · · q−1⊗q0⊗q1· · · qt) =p−n· · · p−1⊗p0⊗p1· · · pmq−s· · · qt, (p−n· · · p−1⊗p0⊗p1· · · pm)`(q−s· · · q−1⊗q0⊗q1· · · qt) =p−n· · · pmq−s· · · q−1⊗q0⊗q1· · · qt, where pi, qi∈p. Note that central dots in the previous expressions represent tensor products, but they are omitted in order to indicate clearly the middle entry, which is p0in the first case and q0in the second one. Given a Leibniz homomorphism f:p→m, we can define Ud(f): Ud(p)→Ud(m) by extending the definition of fto the elements of Ud(p) by linearity. In this section we extend the functors Lb and Udto the categories XDias and XLb in such a way that those extensions are still adjoint to one another. 3.3.1 From XDias to XLb Let us first show that the functor Lb preserves the semidirect product. Lemma 3.3.1. Given Dand Ltwo associative dialgebras together with an action of Don L, there is an action of Lb(D)on Lb(L)given by [x, a] = xaa−a`x, [a, x] = aax−x`a, for all x∈Lb(D),a∈Lb(L). Proof. The six conditions from the Definition 1.2.37 can be checked by straightforward calculations, using the 30 equalities from the Definition 1.2.50. We show here how to check the first condition; the rest of them can be done analogously. Let a∈Lb(L), x, x0∈Lb(D). We will prove that [a, [x, x0]] = [[a, x], x0]−[[a, x0], x]. [a, [x, x0]] = [a, x ax0−x0`x] = aa(xax0−x0`x)−(xax0−x0`x)`a =aa(xax0) | {z } (2) −aa(x0`x) | {z } (1) −(xax0)`a | {z } (4) +(x0`x)`a | {z } (5) , [[a, x], x0]=[aax−x`a, x0] = (aax−x`a)ax0−x0`(aax−x`a) = (aax)ax0 | {z } (2) −(x`a)ax0 | {z } (3) −x0`(aax) | {z } (30) +x0`(x`a) | {z } (5) , −[[a, x0], x] = −(aax0)ax |{z } (1) +(x0`a)ax | {z } (30) +x`(aax0) | {z } (3) −x`(x0`a) | {z } (4) . 118 3 Adjunctions between categories of crossed modules The addends labelled by (1) are equal because of the equality in Definition 1.2.50 constructed from (Di1), with the first element in Land the other two in D. The same happens with the parts labelled by (2), (4) and (5), using the corresponding equalities from (Di2), (Di4) and (Di5). The parts labelled by (3) cancel each other due to the equality in Definition 1.2.50 constructed from (Di3) with the first and the third elements in Dand the second in L. The same applies to (30). Lemma 3.3.2. Let Dand Lbe two associative dialgebras together with an action of Don L. Then Lb(LoD) = Lb(L)oLb(D). Proof. Due to the previous lemma, Lb(D) acts on Lb(L), so it makes sense to consider the semidirect product Leibniz algebra Lb(L)oLb(D). It is clear that Lb(LoD) and Lb(L)oLb(D) are equal as K-modules, so we only need to check that they share the same bracket. Let (a, x),(a0, x0)∈L×D. If we use the bracket in Lb(L)oLb(D), we get: [(a, x),(a0, x0)] = ([a, a0]+[x, a0]+[a, x0],[x, x0]) = (aaa0−a0`a+xaa0−a0`x+aax0−x0`a, x ax0−x0`x). On the other hand, if we use the bracket in LieAs(BoA), we get [(a, x),(a0, x0)] = (a, x)a(a0, x0)−(a0, x0)`(a, x) = (aaa0+xaa0+aax0, x ax0)−(a0`a+x0`a+a0`x, x0`x), so the brackets are equal. Given a crossed module of dialgebras (L, D, µ), we can now define XLb(L, D, µ) as the Leibniz crossed module (Lb(L),Lb(D),Lb(µ)), with the action of Lb(D) on Lb(L) described in Lemma 3.3.1. Note that Lb(µ) = µ. Directly from (XDi1), (XDi2) and the definition of the Leibniz action, we get that (Lb(L),Lb(D),Lb(µ)) satisfies (XLb1) and (XLb2). Moreover, any morphism of crossed modules of dialgebras (ϕ, ψ): (L, D, µ)→ (L0, D0, µ0) is morphism of crossed modules of Leibniz algebras from (Lb(L),Lb(D), µ) to (Lb(L0),Lb(D0), µ0), since ϕ([x, a]) = ϕ(xaa−a`x) = ψ(x)aϕ(a)−ϕ(a)`ψ(x)=[ψ(x), ϕ(a)], ϕ([a, x]) = ϕ(aax−x`a) = ϕ(a)aψ(x)−ψ(x)`ϕ(a)=[ϕ(a), ψ(x)], for all x∈Lb(D), a∈Lb(L). The previous assignments define a functor XLb :XDias →XLb, which is a natural generalization of the functor Lb :Dias →Lb in the following sense. Recall that we have the full embeddings J0,J1:Lb XLb J0 0,J0 1:Dias XDias 3.3.2 Universal enveloping crossed module of a Leibniz crossed module 119 where J0(p)=({0},p,0), J1(p)=(p,p,idp), J0 0(D)=({0}, D, 0) and J0 1(D) = (D, D, idD). It is obvious that the following diagram is commutative for i= 0,1. Dias XDias Lb XLb . J0 i Lb XLb Ji 3.3.2 Universal enveloping crossed module of a Leibniz crossed module Let us now construct the left adjoint to the functor XUd, which generalizes the universal enveloping dialgebra functor Ud:Lb →Dias to crossed modules. Let (m,p, η) be a Leibniz crossed module and consider its corresponding cat1Leibniz algebra as in the proof of Proposition 1.2.46, that is mop p s t with s(m, p) = pand t(m, p) = η(m) + pfor all (m, p)∈mop. Now, if we apply Ud to the previous diagram, we get Ud(mop)Ud(p). Ud(s) Ud(t) Although it is true that Ud(s)|Ud(p)=Ud(t)|Ud(p)= idUd(p), in general, the second condition for cat1-dialgebras (CDi2) is not satisfied. For instance, if we take m∈ m\ {0}, it is clear that (m, 0) ∈Ker Ud(s) and (m, −η(m)) ∈Ker Ud(t). However, (m, 0) ⊗(m, −ν(m)) 6= 0, so Ker Ud(s)∗Ker Ud(t)6= 0 for ∗=aand ∗=`. Nevertheless, we can consider the quotient Ud(mop) = Ud(mop)/X, where X= Ker Ud(s)aKer Ud(t)+Ker Ud(t)aKer Ud(s)+Ker Ud(s)`Ker Ud(t)+Ker Ud(t)` Ker Ud(s), and the induced morphisms Ud(s) and Ud(t). In this way, the diagram Ud(mop)Ud(p) Ud(s) Ud(t) is clearly a cat1-dialgebra. Note that X⊂Ker Ud(t) and Ud(t)|Ud(p)= idUd(p), since t|p= idp. Given v, w ∈Ud({0}op)≃Ud(p), if v−w∈X, then 0 = Ud(t)(v−w) = v−w. Therefore Ud(p) can be regarded as a subalgebra of Ud(mop). We can now define XUd(m,p, η) as the crossed module of dialgebras given by (Ker Ud(s),Ud(p),Ud(t)|Ker Ud(s)). Sometimes we will write Ud(t) instead of Ud(t)|Ker Ud(s) to ease notation. 120 3 Adjunctions between categories of crossed modules For any morphism of Leibniz crossed modules (ϕ, ψ): (m,p, η)→(m0,p0, η0), XUd(ϕ, ψ) is given by Ker Ud(s)Ud(p) Ker Ud(s0)Ud(p0) Ud(t) Ud(ϕ,ψ)|Ker Ud(s)Ud(ψ) Ud(t0) where Ud(ϕ, ψ) is the algebra homomorphism induced by Ud(ϕ, ψ), which is itself the linear extension of (ϕ, ψ): mop→m0op0, given by (ϕ, ψ)(m, p)=(ϕ(m), ψ(p)) for all (m, p)∈mop. The functor XUd:XLb →XDias is a natural generalization of the functor Ud, in the sense that it makes the following diagram commute, Lb XLb Dias XDias . J0 UdXUd J0 0 Regarding the embeddings J1and J0 1, we have the following result. Proposition 3.3.3. There is a natural isomorphism of functors XUd◦J1∼ =J0 1◦Ud. Proof. Let p∈Lb. We need to show that XUd(p,p,idp) is naturally isomorphic to (Ud(p),Ud(p),idUd(p)). In order to do so, we will prove that (Ud(t)|Ker Ud(s),idUd(p)) is an isomorphism of crossed modules of dialgebras between (Ker Ud(s),Ud(p),Ud(t)|Ker Ud(s)) and (Ud(p),Ud(p),idUd(p)). Note that (Ud(t)|Ker Ud(s),idUd(p)) is indeed a morphism of crossed modules of dialgebras (see Example 1.2.56 (iii)). Recall that the first step in the construction of XUd(p,p,idp) requires us to consider the cat1-Leibniz algebra pop p s t with s(p, p0) = p0and t(p, p0) = p+p0for all p, p0∈p. Let us define the Leibniz homomorphism :p→pop,(p) = (p, 0). It is clear that s = 0 and t = idp. The next step is to apply the functor Udon the previous cat1-Leibniz algebra and take the quotient of Ud(pop) by X= Ker Ud(s)aKer Ud(t) + Ker Ud(t)aKer Ud(s) + 3.3.2 Universal enveloping crossed module of a Leibniz crossed module 121 Ker Ud(s)`Ker Ud(t) + Ker Ud(t)`Ker Ud(s) in order to guarantee that we have a cat1-dialgebra. In the next diagram of dialgebras, Ud(p)Ud(pop)Ud(p) Ud(pop)/X Ud() π Ud(s) Ud(t) Ud(s) Ud(t) where πis the canonical projection, it is easy to see that Ud(s)πU() = Ud(s)Ud() = Ud(s) = 0 and Ud(t)πUd() = Ud(t)Ud() = Ud(t) = idUd(p). Hence πUd() takes values in Ker Ud(s) and it is a right inverse for Ud(t)|Ker Ud(s). Now we need to show that πUd()Ud(t) = idKer Ud(s). Note that X⊂Ker Ud(s), so Ker Ud(s) = Ker Ud(s)/X and, as a K-module, Ker Ud(s) is generated by all the elements of the form (p−n, p0 −n)⊗ · · · ⊗ (pi,0) ⊗ · · · ⊗ (pm, p0 m) (3.3.1) with n, m ∈N,pi, p0 i∈p,−n≤i≤m. By the definition of Ud(t) and Ud(), the value of Ud()Ud(t) on (3.3.1) is (p−n+p0 −n,0) ⊗ · · · ⊗ (pi,0) ⊗ · · · ⊗ (pm+p0 m,0).(3.3.2) Furthermore, one can easily derive the following equalities in Ker Ud(s)/X: (p−n+p0 −n,0) ⊗ · · · ⊗ (pi,0) ⊗ · · · ⊗ (pm+p0 m,0) = (p−n, p0 −n)⊗ · · · ⊗ (pi,0) ⊗ · · · ⊗ (pm+p0 m,0) =· · · = (p−n, p0 −n)⊗ · · · ⊗ (pi,0) ⊗ · · · ⊗ (pm, p0 m). Thus, the elements (3.3.1) and (3.3.2) are equal in Ker Ud(s)/X and it follows that πUd()Ud(t)|Ker Ud(s)= idKer Ud(s). Therefore we have found an inverse for the morphism of crossed modules of dialgebras (Ud(t)|Ker Ud(s),idUd(p)), which is given by Ud(p)Ud(p) Ker Ud(s)Ud(p). idUd(p) πUd()idUd(p) Ud(t)