Abelian subalgebras and ideals of maximal dimension in Lie algebras

UNIVERSITY OF SEVILLE Faculty of Mathematics Department of Geometry and Topology ABELIAN SUBALGEBRAS AND IDEALS OF MAXIMAL DIMENSION IN LIE ALGEBRAS Dissertation presented by Manuel Ceballos González to obtain the Ph.D. degree in Mathematics from the University of Seville. Signed: Manuel Ceballos González M.Sc. in Mathematics Seen and approved by the advisors: Juan Núñez Valdés Ángel F. Tenorio Villalón Associate Professor Assistant Professor Dept. Geometry and Topology Dept. Economics, Quantitative Methods, University of Seville and Economic History Pablo de Olavide University Seville, January 2012 ACKNOWLEDGEMENTS It is really dicult to nd the correct words to thank all the people that have been with me through this period of my life. First of all, I should start with my advisors. D. Juan Núñez and D. Ángel Francisco Tenorio are a very important part of this work. I believe that without their suggestions and support in every moment, this dissertation would not exist. I would like to thank D. Juan Núñez for his suggestions and his continuous support to keep improving this work. He has also encouraged me to spend some months abroad during my Ph.D. in order to learn more. Moreover, he was the rst person who introduced me to the mathematical research world. I would like to thank D. Ángel Francisco Tenorio for his patience and the multiple corrections that he has done on this work. He introduced me to the study of abelian subalgebras and ideals and he has also given me a very useful point of view so that we could obtain more and new results. There are other people that have helped me and I would like to dedicate a few words for them. First, Professor Dietrich Burde, from the University of Vienna introduced me to the general concepts of α and β invariants. Then, Professor David Towers from the University of Lancaster, who taught me a lot about supersolvable Lie algebras, maximal subalgebras and other related topics. Finally, I would like to thank Professor Francisco J. Castro from the University of Seville for his useful comments and ideas related to the relations between the coecients in the general laws of liform Lie algebras. Of course, I also have to thank my family, relatives and friends since they have supported me in every moment. There seems to have been a very long process since I started studying Mathematics until this moment that I am writing these words trying to obtain the doctoral degree. Contents Contents iii Introduction v 1 Preliminaries 1 1.1 Denitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Families of solvable Lie algebras . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Lie algebras hn .......................... 7 1.2.2 Lie algebras gn .......................... 8 1.2.3 Heisenberg algebras . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Main and non-main vectors . . . . . . . . . . . . . . . . . . . . . . . 9 2 Theoretical study 11 2.1 Some properties and bounds . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Abelian subalgebras of codimension 1 .................. 15 2.3 Abelian subalgebras of codimension 2 .................. 18 2.4 Abelian subalgebras of codimension 3 .................. 24 3 Abelian ideals and subalgebras in some solvable Lie algebras 29 3.1 Lie algebra gn ............................... 30 3.2 Lie algebra hn ............................... 34 3.3 Heisenbergalgebras............................ 36 3.4 FiliformLiealgebras ........................... 39 4 Algorithm and applications 59 4.1 Algorithmicmethod ........................... 59 4.2 Computation of α and β invariants for low-dimensional Lie algebras . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.1 Lie algebras of dimension less than 5 .............. 74 4.2.2 Solvable non-nilpotent Lie algebras of dimension 5 and 6 . . . 75 iii iv Contents 4.2.3 Nilpotent Lie algebras of dimension less than 8 ......... 78 4.2.4 Tables............................... 82 4.3 Matrix representation of liform Lie algebras . . . . . . . . . . . . . . 97 4.3.1 Model liform Lie algebras . . . . . . . . . . . . . . . . . . . . 98 4.3.2 Non-model liform Lie algebras . . . . . . . . . . . . . . . . . 99 4.3.3 Examples of application . . . . . . . . . . . . . . . . . . . . . 101 4.3.4 Filiform Lie algebras of dimension less than 9 . . . . . . . . . 105 Conclusions 111 Bibliography 113 List of gures 123 List of tables 126 List of authors 127 Introduction Apart from the study of Lie Theory from a purely theoretical view-point, extensive research about this theory exists due to its many applications in Engineering, Physics and, above all, Applied Mathematics. However, some aspects of Lie algebras still remain to be studied. Indeed, the classication of nilpotent and solvable Lie algebras is still an open problem, although the classication of other types of Lie algebras (like semisimple and simple ones) was already obtained in 1890. In order to solve these and other problems, the need of studying additional properties of Lie algebras arises. For example, conditions on the lattice of subalgebras of a Lie algebra often lead to information about the Lie algebra itself. In fact, studying abelian Lie subalgebras and ideals of a nite-dimensional Lie algebra constitutes the main goal of this dissertation. It is also convenient to indicate the motivations leading to obtain more information about Lie algebras in general. In a high percentage, the reasons lie in the possibility of using Lie algebras and their properties as tools in the study of several topics in Physics and Economics, for instance. Indeed, at present, Lie algebras and groups are widely used in Modern Physics. For example, a classic use of Lie Theory corresponds to the study of symmetries (see [55, 87]). Nowadays, symmetries are not limited to those geometrical versions of space-time; but there are other new symmetries associated with internal degrees of freedom of particles and elds. Regarding to possible economic and nance applications, we advise to consult [53] as starting point. Another application of Lie groups and algebras corresponds to Einstein spaces and manifolds. To classify Einstein spaces according to their respective isometry groups in dimension 4, Petrov [85] used Bianchi's works and applied the classication of low dimensional Lie algebras to General Relativity (see also [93]). Besides, a particular type of Einstein solvmanifold (which is called standard) is based on computing an abelian orthogonal complement [51] and, therefore, it would be interesting to obtain an algorithmic method which computes abelian subalgebras in a given non-abelian Lie algebra. v vi Introduction Let us consider a nite-dimensional Lie algebra, g . Let α(g) denote the maximal dimension of an abelian subalgebra of g , and β(g) the maximal dimension of an abelian ideal of g . Both invariants are important for many subjects. First of all they are very useful in the study of Lie algebra contractions and degenerations. There is extensive literature on these topics, in particular for low-dimensional Lie algebras, as can be seen in [19, 48, 49, 77, 92] and the references given therein. The rst author dealing with the invariant α(g) was Schur [91], who studied in 1905 the abelian subalgebras of maximal dimension contained in the Lie algebra of n×n square matrices. Schur proved that the maximum number of linearly independent commuting n×n matrices over an algebraically closed eld is [n2 4]+ 1 , which is the maximal dimension of abelian ideals of Borel subalgebras in the general linear Lie algebra gl(n) (where [x] denotes the integer part of a real number x ). Let us note that this result was obtained only over algebraically closed elds such as the complex number eld. Almost forty years later, in 1944, Jacobson [57] gave a simpler proof of Schur's results, extending them from algebraically closed elds to arbitrary elds. This fact allowed several authors to deal later with the study of the abelian subalgebras of maximal dimension of many dierent types of Lie algebras. As it is well-known, Lie algebras can be distinguished in three dierent types: solvable ones, semisimple ones and the remaining algebras which do not belong to any of the two previous types. More concretely, Levi [69] and Malcev [72] proved (in 1905 and 1945, respectively) that every nite-dimensional Lie algebra can be expressed as a semi-direct sum of a semisimple subalgebra and its radical (i.e. a solvable ideal). Regarding this, it was also proved that every semisimple Lie algebra can be decomposed in a direct sum of simple Lie algebras. Hence, the classication of Lie algebras can be reduced to obtain the classication of both semisimple and solvable Lie algebras. Additionally, Killing [59, 60, 61, 62] and Cartan [20] (among others) obtained the classication of simple Lie algebras at the end of the nineteenth century. Moreover, taking advantage of Killing and Cartan's classication, Malcev [72] computed the α invariant for semisimple Lie algebras in 1945 giving the value of this invariant for simple ones according to Table 1. Since there are no abelian ideals in a simple Lie algebra s , we have β(s) = 0 . Very recently, the study of abelian ideals in a Borel subalgebra b of a simple complex Lie algebra s has drawn considerable attention. In this case, α(s) = β(b) holds, and this value can be computed purely in terms of certain root system invariants, see [97]. Konstant [66] also dealt with the topic of abelian ideals in a Borel subalgebra of a given Lie algebra g by studying the proof of the following theorem, which has been Introduction vii Table 1: The invariant α for simple Lie algebras sdim(s)α(s) An, n ≥1n(n+ 2) ⌊(n+1 2)2⌋ B321 5 Bn, n ≥4n(2n+ 1) n(n−1) 2+ 1 Cn, n ≥2n(2n+ 1) n(n+1) 2 Dn, n ≥4n(2n−1) n(n−1) 2 G214 3 F452 9 E678 16 E7133 27 E8248 36 attributed to Peterson in [66]: the number of abelian ideals in a xed Borel subalgebra of g is 2r , where r= rank(g) . Regarding this, some authors like Cellini, Papi and Orsina [30, 80] have recently obtained new properties of the maximal abelian ideals in a Borel subalgebra as well as a generalization of Peterson's theorem from abelian ideals to ad-nilpotent ones. Furthermore, Kostant [66, 67] found a relation of these invariants α and β with discrete series representations of the corresponding Lie group, and with powers of the Euler product. In fact, there are much more results concerning the invariants α and β for simple Lie algebras and their Borel subalgebras. There are also several results concerning the question of how large or small these maximal dimensions can be, in comparison with the dimension of the Lie algebra (see [73, 88, 94] for example). The results show, roughly speaking, that a Lie algebra of large dimension contains abelian subalgebras of large dimension. For example, the dimension of a nilpotent Lie algebra g satisfying α(g) = ℓ is bounded by dim(g)≤ℓ(ℓ+1) 2 [88, 94]. There is a better bound in [74] for 2 -step nilpotent Lie algebras with α(g) = l and ℓ≥8 : dim(g)≤[ℓ2+4 8] + ℓ . If g is a complex solvable Lie algebra with α(g) = ℓ , then the bound dim(g)≤ℓ(ℓ+3) 2 was proved in [73]. In general, dim(g)≤ℓ(ℓ+17) 2 for any complex Lie algebra g with α(g) = ℓ , see [73]. The problem of classifying abelian subalgebras of Lie algebras also has a long history, starting from the Killing-Cartan classication of semisimple Lie algebras. However, the problem of determining such subalgebras is far from being solved, due to the lack of structural criteria for general types of Lie algebras. xiv Introduction independently of using associative algebras or Lie ones. After this, she determined both upper and lower bounds of the function for the Lie case, by using the previously referred lemma and the results already obtained in [73]. The proof concluded when she obtained that upper and lower bounds were the same. After having commented this brief historical view on Lie algebras in general and on abelian Lie subalgebras in particular, now we will deal with this dissertation. Its structure is the following: Chapter 1 recalls those more general concepts on Lie algebras which will be cited throughout this dissertation. Chapter 2 constitutes a theoretical study of abelian subalgebras and ideals contained in Lie algebras. First, we give some general properties and bounds. More concretely, we want to point out an interesting result for solvable Lie algebras: if g is a solvable Lie algebra over an algebraically closed eld of characteristic zero, then α(g) = β(g) . This means that, given an abelian subalgebra of maximal dimension m there exists also an abelian ideal of dimension m . For a given value of α(g) , the dimension of g is bounded in terms of this value, as mentioned above. It is natural to ask what we can say on an n -dimensional Lie algebra g involving the value of α(g) is close to n . Indeed, if α(g) = n , then g is abelian and α(g) = β(g) . If α(g) = n−1 , then also β(g) = n−1 . This means that g has an abelian ideal of codimension 1 and is almost abelian. In particular, g is 2 -step solvable. In this case the structure of g , and even all its degenerations are quite well-understood, see [48]. These two easy cases suggest to consider Lie algebras g satisfying α(g) = n−2 . Here we can classify all such nonsolvable Lie algebras. For the solvable case, we characterize these algebras and prove that every supersolvable Lie algebra with an abelian subalgebra of codimension 2 has also an abelian ideal with the same dimension. We also give a method to obtain an abelian ideal of codimension 2 from an abelian subalgebra of the same dimension in a nilpotent Lie algebra. Let us note that for many problems concerning the cohomology of nilpotent Lie algebras, the subclass of those having an abelian ideal of codimension 1 or 2 is very important, as can be seen in [2, 86] and the references given therein. In order to conclude the chapter, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying eld is not two. We also give several examples to clarify some results. Chapter 3 is devoted to show several algorithmic methods and results about the abelian subalgebras of maximal dimension for the most important families of solvable Lie algebras: Lie algebra gn , of n×n strictly upper-triangular matrices, Lie algebra hn , given by n×n upper-triangular matrices, Heisenberg algebras and liform Lie algebras. First, we show a method to compute an abelian subalgebra of Introduction xv maximal dimension for Lie algebras gn , hn and Heisenberg algebras. This procedure is dened as follows: given a Lie algebra, g , we consider an arbitrary basis of an r -dimensional subalgebra with respect to the basis Bg of the Lie algebra g in which its law is expressed. Then the vectors in Bg are divided in main vectors or nonmain vectors according to a echelon matrix associated with Bg . Finally it is studied whether the subalgebra is abelian or not. When only non-abelian subalgebras are obtained, we have proved that the α invariant of g is less than r . The main reason why we have focused on the Lie algebras hn and gn is that every nite-dimensional solvable Lie algebra can be represented by a Lie subalgebra of some hn (see [43, Theorem 9.11] or [105, Theorem 3.7.3]) and every nite-dimensional nilpotent Lie algebra is isomorphic to a subalgebra of gn ([105, Proposition 3.6.6]). Another reason is that its applications to Physic are many and varied (e.g. [54, 87]). In order to conclude the chapter, we study the abelian subalgebras of maximal dimension for liform Lie algebras. We prove that there exists a unique abelian ideal of maximal dimension for these algebras and we give several results concerning the description of the general law of a liform Lie algebras by using several invariants. Finally, we give an algorithmic procedure which computes the law of an n -dimensional non-model liform Lie algebra g starting from the value of α(g) . In Chapter 4 , we show an algorithmic method to compute abelian subalgebras and ideals of any nite-dimensional Lie algebra, starting from the non-zero brackets in its law. To implement this algorithm we use the symbolic computation package Maple 12 . Additionally, we give a brief computational study considering both the computing time and the memory used in the two main routines of the implementation. We have also studied the complexity and number of operations. Moreover, we have included in this chapter two dierent applications of the previous results. The rst is related to the computation of the α and β invariants for low dimensional Lie algebras. More concretely, we compute these invariants for Lie algebras of dimension less than 5 in general, solvable Lie algebras of dimension less than 7 and nilpotent Lie algebras of dimension less than 8 . The second application consists in the study of minimal faithful unitriangular matrix representations for liform Lie algebras. xvi Introduction Spanish summary En esta sección exponemos un resumen en español del contenido de este trabajo, en el que se estudian dos invariantes de las álgebras de Lie: las dimensiones maximales de sus subálgebras e ideales abelianos. Así, si g es un álgebra de Lie de dimensión nita, denotamos por α(g) y β(g) al máximo entre la dimensión de todas las subálgebras e ideales abelianos de g , respectivamente. Estos invariantes fueron introducidos por Schur en 1905 al estudiar las subálgebras abelianas maximales en el álgebra formada por matrices cuadradas. Se usan en el estudio de contracciones y degeneraciones de álgebras de Lie. Hay muchos artículos tratando estos temas, por ejemplo: [49, 19, 77, 92, 48]. Asimismo, algunos resultados comparan el valor de estos invariantes con la dimensión de la propia álgebra ([94, 88, 73]). En el Capítulo 1 exponemos conceptos preliminares necesarios para la adecuada comprensión del resto del trabajo. A continuación, en el Capítulo 2 , se analizan propiedades básicas y generales de los invariantes α y β , como son la aditividad y monotonía, y diversas cotas. También se exponen una serie de resultados comparando ambos invariantes. En concreto, se demuestra que si g un álgebra de Lie de dimensión nita y se consideran su descomposición de Levi-Malcev g=snr , entonces α(snr)≤α(s⊕r) . También se prueba que si g es un álgebra de Lie resoluble sobre un cuerpo algebraicamente cerrado de característica cero, entonces ambos invariantes coinciden. A continuación, se estudian dos casos particulares correspondientes a subálgebras abelianas de codimensión 1 y 2. Para el primer caso, se prueba que si g un álgebra de Lie de dimensión n vericando α(g) = n−1 , entonces β(g) = n−1 . Las álgebras de Lie vericando β(g) = n−1 son álgebras de Lie casi abelianas, que son álgebras de Lie resolubles 2 -step. En el caso de codimensión 2 , también podemos dar una caracterización: Si g es un álgebra de Lie de dimensión n vericando α(g) = n−2 , entonces, o bien g es isomorfa a sl2(C)⊕Cℓ o bien es resoluble. Además damos una caracterización en ambos casos. También se prueba que toda álgebra de Lie nilpotente con una subálgebra abeliana de codimensión 2, contiene un ideal abeliano de la misma dimensión. Notemos que los ideales abelianos de álgebras de Lie nilpotentes de codimensión 1 y 2 juegan un papel importante en problemas de cohomología de estas álgebras (ver [2, 86]). Por último, estudiamos el caso de codimensión 3 para las álgebras nilpotentes sobre un cuerpo de característica distinta de dos en el que se muestra un método para obtener un ideal abelian de codimensión 3 a partir de una subálgebra abeliana de la misma dimensión. En el tercer capítulo, se analizan las subálgebras e ideales abelianos en varias familias de álgebras de Lie resolubles. En concreto, estudiamos las álgebras de Lie gn , dadas por matrices cuadradas estrictamente triangulares superiores, las álgebras hn , Spanish summary xvii dadas por matrices cuadradas triangulares superiores y las álgebras de Heisenberg. La principal razón por la que hemos estudiado estas familias es que cada álgebra de Lie resoluble de dimensión nita se puede representar por una subálgebra de hn (ver [43, Theorem 9.11] o [105, Theorem 3.7.3]) y cada álgebra de Lie nilpotente de dimensión nita es isomorfa a una subálgebra de gn ([105, Proposition 3.6.6]). Otra razón es por sus múltiples aplicaciones en Física (por ejemplo [54, 87]). Para concluir este capítulo, estudiamos las álgebras de Lie liformes. Se prueba que para estas álgebras existe un único ideal abeliano de máxima dimensión. También damos algunos resultados sobre la descripción de la ley general de estas álgebras a paritr de varios invariantes. Finalmente damos un procedimiento algorítmico para calcular la ley general de un álgebra de Lie liforme a partir del valor del invariante alfa. En el Capítulo 4 , mostramos un método algorítmico para calcular subálgebras e ideales abelianos de un álgebra de Lie arbitraria a partir de su ley. Para implementar este procedimiento, hemos utilizado el programa de computación simbólica Maple 12 . Además, damos un breve estudio computacional de las principales rutinas analizando el tiempo computacional y la memoria usada. Se adjunta también una tabla con la complejidad computacional y el número de operaciones para estas rutinas. A continuación, y para nalizar este capítulo, hemos incluido dos aplicaciones correspondientes al cálculo del invariante α para álgebras de Lie de pequeña dimensión y la representación matricial de álgebras de Lie liformes. Más concretamente, estudiamos el valor de α para álgebras de Lie de dimensión menor que 5 de cualquier tipo, álgebras de Lie resolubles de dimensión menor que 7 y nilpotentes de dimensión menor que 8 . Para las representaciones matriciales de álgebras de Lie liformes, se consideran las álgebras gn y sus subálgebras abelianas calculadas en el capítulo anterior. Mostramos la representación general de las álgebras de Lie liformes modelos, un método para el cálculo de la representación para las no modelos y, como ejemplo, varias tablas con la representación de álgebras de Lie liformes de dimensión menor que 9 . Chapter 1 Preliminaries This chapter is devoted to recall some preliminary concepts and results on Lie algebras. For a general overview, the interested reader can consult [105]. Throughout this dissertation, g will denote a nite-dimensional Lie algebra over a eld K . The assumptions on K will be specied in each section or result. 1.1 Denitions and notations Denition 1.1. A Lie algebra g over an arbitrary eld K is a vector space over K endowed with a second inner law, named the bracket product , and verifying the following three properties 1. Bilinearity: [αu +βv, w] = α[u, w] + β[v, w],[u, αv +βw] = α[u, v] + β[u, w] , for α, β ∈K,∀u, v ∈g . 2. [u, u] = 0,∀u∈g . 3. Jacobi Identity: [[u, v], w] + [[v, w], u] + [[w, u], v] = 0,∀u, v, w ∈g . Denition 1.2. The dimension of a Lie algebra is its dimension as a vector space. Let us consider a Lie algebra g with a basis {ei}n i=1 . Such a basis can be characterized by the structure constants (or Maurer-Cartan constants ), dened by [ei, ej] = ∑ch i,jeh for any 1≤i<j≤n . These constants determine the whole structure of the Lie algebra. Remark 1.1. The second condition in Denition 1.1, together with the bilinearity of the bracket product, implies the skew-symmetry over a eld of characteristic dierent from 2 ; i.e. [u, v] = −[v, u] , for all u, v ∈g . 1 2 Chapter 1. Preliminaries Denition 1.3. Given a Lie algebra g , a vector subspace h of g is a subalgebra if [u, v]∈h , for all u, v ∈h . Moreover, the subalgebra h is an ideal of g if [h1, g1]∈h , for all h1∈h and for all g1∈g . Denition 1.4. Given a subalgebra h of a Lie algebra g , the core of h , denoted by hg , is the largest ideal of g contained in h . Denition 1.5. Given a subalgebra h of a Lie algebra g , the ideal closure of h , denoted by hg , is the smallest ideal of g containing h . Denition 1.6. Let g be a nite-dimensional Lie algebra and h⊆g a subalgebra. The centralizer of h in g is dened as Cg(h) = {g∈g|[g, h] = 0,∀h∈h} . Denition 1.7. Let g be a nite-dimensional Lie algebra and h⊆g a subalgebra. We dene the normalizer of h in g as Ng(h) = {g∈g|[g, h]∈h,∀h∈h} . Denition 1.8. Given an element x∈g , the adjoint endomorphism or adjoint action is a Lie-algebra endomorphism adx:g→g dened as y∈g7→ adx(y) = [x, y] . Denition 1.9. A derivation on a Lie algebra g is a linear map D:g→g verifying D([x, y]) = [D(x), y]+[x, D(y)]. The set of all derivations of g will be denoted by Der(g) . Let us note that adx is a derivation for all x∈g and ad :g→Der(g) . Denition 1.10. The Frattini subalgebra F(g) of a Lie algebra g is dened as the intersection of all maximal subalgebras of g . The Frattini ideal ϕ(g) is the largest ideal contained in F(g) . Denition 1.11. The linear adjoint group of G , AdG , is the image of the Lie group or algebraic group G under the adjoint representation. The adjoint group AdG is contained in the group Aut(g) of automorphisms of the Lie algebra g of G and its Lie algebra coincides with the adjoint algebra ad(g) of g . Theorem 1.1 (Borel xed-point theorem in Borel [9]) . Let G be a connected, solvable algebraic group acting regularly on a non-empty, complete algebraic variety V over an algebraically closed eld K . Then G has a xed point in V . There exist three dierent types of Lie algebras: solvable algebras, semisimple ones and those which do not belong to these two previous types, but can be expressed as a semidirect sum of two algebras of the previous types. Denitions and notations 3 Denition 1.12. A Lie algebra g over a a eld of characteristic zero is semisimple if g is not abelian and does not contain any non-zero proper abelian ideal. In addition, the Lie algebra g is said to be simple if it is not abelian and does not contain any non-zero proper ideal. Denition 1.13. Let g be a nite-dimensional Lie algebra. The upper central series or derived series of g is dened by C1(g) = g,C2(g) = [g,g],C3(g) = [C2(g),C2(g)], . . . , Ck(g) = [Ck−1(g),Ck−1(g)], . . . If there exists m∈N such that Cm(g)≡0 , the Lie algebra g is called solvable . Moreover, g is (m−1) -step solvable if there exists m∈N such that Cm(g)≡ {0} and Cm−1(g)={0} . A special class of solvable Lie algebras is formed by abelian algebras. Denition 1.14. A Lie algebra g is abelian if [v, w] = 0 , for all v , w∈g . A very important and interesting abelian subalgebra (in fact, an abelian ideal) in a given Lie algebra g is the center of the algebra. Denition 1.15. The center Z(g) of a Lie algebra g , is dened as follows Z(g) = {u∈g|[u, v] = 0,∀v∈g} . Denition 1.16. The lower central series or simply central series of a Lie algebra g is dened by C1(g) = g,C2(g) = [C1(g),g],C3(g) = [C2(g),g], . . . , Ck(g) = [Ck−1(g),g], . . . If there exists m∈N such that Cm(g)≡0 , the Lie algebra g is called nilpotent . We will say that g is (m−1) -step nilpotent if there exists m∈N such that Cm(g)≡ {0} and Cm−1(g)={0} . Denition 1.17. The derived Lie algebra D(g) of a Lie algebra g is given by D(g) = C2(g) = C2(g) . Remark 1.2. Lie algebras with its derived algebra being abelian correspond to 2 -step solvable ones and are usually called metabelian . Remark 1.3. Let us note that every nilpotent Lie algebra is also solvable, because Ci(g)⊆ Ci(g),∀i . Related to the lower central series associated with a subalgebra of g , the following result holds 4 Chapter 1. Preliminaries Proposition 1.1. Let h be a subalgebra of a Lie algebra g . Then Ck(h)⊆ Ck(g) , for all k∈N . Denition 1.18. The radical of a Lie algebra g is dened as its maximal solvable ideal and the nilradical is its maximal nilpotent ideal. Denition 1.19. A Borel subalgebra of a Lie algebra g is a maximal solvable subalgebra of g . Denition 1.20. A Cartan subalgebra of g is a nilpotent subalgebra which is selfnormalizing. The rank of g is given by the dimension of its Cartan subalgebras. Denition 1.21. A Lie algebra g is supersolvable if there is a chain 0 = I0⊂I1⊂ . . . ⊂In−1⊂In=g , where Ik is a k -dimensional ideal of g Remark 1.4. It is well-known that every supersolvable Lie algebra is also solvable. Moreover, these classes coincide over an algebraically closed eld of characteristic zero (Lie's Theorem [105]). There are, however, examples of solvable Lie algebras over algebraically closed eld of non-zero characteristic which are not supersolvable, as can be seen in [5] or [58, page 53]. Denition 1.22. A subalgebra h of g is said to be triangulable on g if adgh= {adgx|x∈h} is a Lie algebra of linear transformations of g which is triangulable over the algebraic closure of K . Proposition 1.2 (Theorem 2.2 in [109]) . A subalgebra h of g is triangulable on g if and only if every element of D(h) acts nilpotently on g . Denition 1.23. Let us consider a vector space V and an endomorphism f over it. Then, V can be decomposed into the direct sum of two subspaces which are invariants under the action of f as V=V0⊕V1, where f|V0 is nilpotent and f|V1 is an isomorphism. The denition of these subspaces is as follows V1=∞ ∩ i=1 fi(V) and V0={v∈V| ∃r∈N, frv= 0}. This is known as the Fitting decomposition . Denition 1.24. A nilpotent Lie algebra g is said to be liform if it veries that dim(C2(g)) = n−2; . . . dim(Ck(g)) = n−k;. . . dim(Cn(g)) = 0, where dim(g) = n . Denitions and notations 5 A basis {ei}n i=1 of g is called adapted if [e1, e2] = 0,[e1, eh] = eh−1(3 ≤h≤n), [e2, eh] = 0 (3 ≤h≤n),[e3, eh] = 0 (4 ≤h≤n). Remark 1.5. Note that the denition of liformity assures that every liform Lie algebra has an adapted basis, as it was proved in [107]. Remark 1.6. It is easy to deduce that, with respect to adapted bases, the following conditions hold. C2(g)≡ {ei}n−1 i=2 ,C3(g)≡ {ei}n−2 i=2 , . . . , Cn−1(g)≡ {e2},Cn(g)≡ {0}. Proposition 1.3. Let g be a n -dimensional liform Lie algebra with an adapted basis {ei}n i=1 , then it is veried that [Cp(g),Cq(g)] ⊂ Cmin{p,q}+1(g) . Proof. Let us note that if p < q then Cq(g)⊂ Cp(g) . Therefore, [Cp(g),Cq(g)] ⊂ [Cp(g),Cp(g)] ⊂[Cp(g),g] = Cp+1(g) . Denition 1.25. A liform Lie algebra g is said to be model if the only nonzero brackets between the elements of an adapted basis are: [e1, eh] = eh−1 , for h= 3, . . . , n . In [34] (although by using a dierent notation, which was later improved in [35]), the following invariants were introduced Denition 1.26. Let g be an n -dimensional complex liform Lie algebra. Then, the following invariants of g are dened z1=z1(g) = max{k∈N| Cg(Cn−k+2(g)) ⊃ C2(g)}, z2=z2(g) = max{k∈N|Cn−k+1(g) is abelian }. Remark 1.7. The denition of z1 means that Cn−z1+2(g) is the largest ideal of g whose centralizer contains C2(g) ; i.e., the ideal whose centralizer is the ideal g , generated by {ei}n−1 i=2 with respect to an adapted basis {ei}n i=1 In addition, z1(g) is an invariant for non-model liform Lie algebras. In terms of an adapted basis, it is deduced in [34] that it can be written as follows z1=min{k∈N−{1} | [ek, en]= 0}. and [eh, ek] = 0,for 1 < h < z1and k > 1. (1.1) 12 Chapter 2. Theoretical Study maximal dimension of a commutative subalgebra of s , i.e., α(s) = β(b) . Furthermore the number of abelian ideals in b is 2rank(s) . This implies α(b) = β(b) , because we have α(b)≤α(s) = β(b) , since α is monotone. Lemma 2.1. The invariant α is monotone and additive: for a subalgebra h⊆g , α(h)≤α(g) holds; and for two Lie algebras a and b , α(a⊕b) = α(a) + α(b) . The invariant β need not be monotone. For example, consider a Cartan subalgebra h in g=sl2(C) . Then β(h) = 1 >0 = β(g) . If K is the algebraic closure of K we put h=h⊗KK for every subalgebra h of g . We have the following Lemma 2.2. It is veried that α(g)≥α(g) and β(g)≥β(g) . Considering the Levi decomposition, we can set the following result. Lemma 2.3. Let g be a complex Lie algebra with a Levi decomposition g=snr . Then, α(snr)≤α(s) + α(r) . Proof. Let a be an abelian subalgebra in g of maximal dimension and π:snr−→ s,(x, a)−→ (x, 0) be the projection. Restricting this Lie algebra homomorphism to a yields dim(a) = dim(ker(πa)) + dim(im(πa)) . Since im(πa) is the homomorphic image of a subalgebra of snr , we can assume that im(πa) is an abelian subalgebra of s . In particular we have dim(im(πa)) ≤α(s) . Furthermore we have ker(πa) = a∩r . Hence ker(πa) is an abelian subalgebra of r and we have dim(ker(πa)) ≤α(r) . Finally we obtain α(snr) = dim(a) = dim(ker(πa))+dim(im(πa)) ≤α(s)+α(r). We will also need the following lemma. Lemma 2.4. The center Z(g) of g is contained in any abelian subalgebra of maximal dimension. Proof. We know that an abelian subalgebra a of maximal dimension is self-centralizing, i.e., a=Cg(a) = {x∈g|[x, a] = 0} . Since Z(g)⊂Cg(a) , the claim follows. The fact that α(b) = β(b) for a Borel subalgebra b of a complex simple Lie algebra can be generalized to all solvable Lie algebras. Proposition 2.2. Let g be a solvable Lie algebra over an algebraically closed eld K of characteristic zero. Then, β(g) = α(g) . Some properties and bounds 13 Proof. The result follows easily from the proof of Theorem 4.1 of [38]. For the convenience of the reader we give the details. Let G be the adjoint algebraic group of g . This is the smallest algebraic subgroup of the automorphisms of g , Aut(g) , such that its Lie algebra Lie(G) contains ad(g) . Then Lie(G) is the algebraic hull of ad(g) . Since ad(g) is solvable, so is Lie(G) . Therefore G is a connected solvable algebraic group. Let us suppose that m=α(g) . Consider the set C of all abelian subalgebras of g of dimension m . This is, by assumtion, a non-empty set, which can be considered as a subset of the Grassmannian Gr(g, m) , which is an irreducible complete algebraic variety. Hence, C is a non-empty complete variety, and G operates morphically on it, mapping each commutative subalgebra h on g(h) , for g∈G . By Borel's xed point theorem, G has a xed point I in C , i.e., a subalgebra I of g with g(I) = I for all g∈G . In particular we have ad(x)(I) = I for all x∈g . Hence I is an abelian ideal of dimension m of g . Borel's xed point theorem relies on the closed orbit lemma. As a corollary one can also obtain the Lie-Kolchin theorem [64]. We note that the assumption on K is really necessary. Example 2.2. Let g be the solvable Lie algebra of dimension 4 over R dened by [x1, x2] = x2−x3,[x1, x4] = 2x4, [x1, x3] = x2+x3,[x2, x3] = x4. We prove that, over R , we have α(g) = 2 , but β(g) = 1 . Let K be equal to R or C . Obviously, ⟨x3, x4⟩ is an abelian subalgebra of dimension 2 over K . Assume that α(g)=3 . Then g is almost abelian, hence 2 -step solvable. This is impossible, as g is 3 -step solvable. Hence α(g) = 2 over K . Assume that I is a 2 -dimensional abelian ideal over K . It is easy to see that we can represent I as ⟨ax2+bx3, x4⟩ with a, b ∈K . Obviously both x2 and x3 cannot belong to I . Hence a= 0 and b= 0 . We have ax2+bx3∈I and [x1, ax2+bx3] = (a+b)x2−(a−b)x3∈I . This implies a2+b2= 0 . This is a contradiction over R , so that β(g)=1 in this case. Over C we may take a= 1 and b=i , and I=⟨x2+ix3, x4⟩ is a 2 -dimensional abelian ideal. In Section 2.4, we will see an example concerning the fact that the assumption on the characteristic of the eld K is really necessary. As a consequence, we have the following bounds for α invariant. 14 Chapter 2. Theoretical Study Lemma 2.5. Let g be a complex, non-abelian, nilpotent Lie algebra of dimension n . Then √8n+ 1 −1 2≤α(g)≤n−1 Proof. The estimate is well-known for β(g) , see [38]. By Proposition 2.2 , it follows for α(g) . Lemma 2.6. Let g be any solvable Lie algebra with nilradical N . Then Cg(N)⊆N . Proof. Suppose that Cg(N)⊆ N . Then there is a non-trivial abelian ideal I/(N∩ Cg(N)) of g/(N∩Cg(N)) inside Cg(N)/(N∩Cg(N)) . But now C3(I)⊆[I, N] = 0 , so I is a nilpotent ideal of g . It follows that I⊆N∩Cg(N) , which is a contradiction. Next, we have a bound of β(g) for certain metabelian Lie algebras. Proposition 2.3. Let g be a metabelian Lie algebra of dimension n , and suppose that dim(D(g)) = k . Then dim(g/Cg(D(g))) ≤[k2/4] + 1 . If, further, g splits over D(g) , then β(g)≥n−[k2/4] −1 . Proof. Let ad :g→ Der D(g) be dened by ad x(y) = [y, x] for all y∈ D(g) . Then ad is a homomorphism with kernel Cg(D(g)) . It follows that g/Cg(D(g)) ∼ =D where D is an abelian subalgebra of Der D(g)∼ =gl(k, K) . It follows from Schur's Theorem on commuting matrices (see [58]) that dim(g/Cg(D(g)) ≤[k2/4] + 1 . Now suppose that g=D(g)⊕B , where B is an abelian subalgebra of g . Then Cg(D(g)) = D(g)⊕B∩Cg(D(g)) which is an abelian ideal of g . Now, we obtain bounds for supersolvable Lie algebras by following a development similar to [94, Lemma 2]. Lemma 2.7. Let g be a supersolvable Lie algebra and let I be a maximal abelian ideal of g . Then, Cg(I) = I . Proof. We have that Cg(I) is an ideal of g . Suppose that Cg(I)=I . Let J/I be a minimal ideal of g/I with J⊂ Cg(I) . Then, for some j∈J , J=I+⟨j⟩ , which is an abelian ideal of g , contradicting the maximality of I . Proposition 2.4. Let g be a supersolvable Lie algebra and let I be any maximal abelian ideal of g . Suppose that dim(I)≤k . Then, g/I is isomorphic to a Lie algebra of k×k lower triangular matrices. Theoretical Study 15 Proof. Let ad :g→Der(I) be dened by ad x(y) = [y, x] for all x∈g and y∈I . Then, ad is a homomorphism with kernel Cg(I) = I by Lemma 2.7. Since g is supersolvable there is a ag of ideals 0 = I0⊂I1⊂. . . ⊂Ik=I of g . Choose a basis {ei}k i=1 for I with ei∈Ii . With respect to this basis the action of g on I is represented by k×k lower triangular matrices, since [Ii,g]⊆Ii for each 0≤i≤k . Corollary 2.1. Let g be a supersolvable Lie algebra with a maximal abelian ideal I of dimension at most k . Then dim(g)≤k(k+3) 2 and g has derived length at most k+ 1 . Proof. The Lie algebra of k×k lower triangular matrices has dimension k(k+1) 2 and derived length k . Corollary 2.2. If g is supersolvable of dimension n , then β(g)≥⌊√8n+ 9 −3 2⌋. Corollary 2.3. If g is solvable, but non-abelian of dimension n over an algebraically closed eld K of characteristic zero, then ⌊√8n+ 9 −3 2⌋≤α(g)≤n−1. Proof. Simply use Corollary 2.2 and Proposition 2.2. 2.2 Abelian subalgebras of codimension 1 Let g be a Lie algebra satisfying α(g) = n−1 . We will show that β(g) = n−1 without using Proposition 2.2 . Our proof will be constructive. We do not only show the existence of an abelian ideal of dimension n−1 , but really construct such an ideal from a given abelian subalgebra of dimension n−1 . Note that Lie algebras g with β(g) = n−1 are called almost abelian . As mentioned before, they are 2 -step solvable, and their structure is well-known (see [48, Section 3 ] and [98]). Proposition 2.5. Let g be an n -dimensional Lie algebra satisfying α(g) = n−1 . Then we have β(g) = n−1 , and g is almost abelian. 16 Chapter 2. Theoretical Study Proof. Let a be an abelian subalgebra of dimension n−1 . If D(g)⊆a , then a is also an abelian ideal, and we are done. Otherwise we choose a basis {ei}n i=1 of g such that a=⟨e2, . . . , en⟩ . We have [ej, eℓ] = 0 for all j, ℓ ≥2 . There exists k≥2 such that [e1, ek] is not contained in a . We may assume that k= 2 by relabelling e2 and ek . For j≥2 , let us consider [e1, ej] = αj,1e1+αj,2e2+···+αj,nen. We have α2,1= 0 . Rescaling e1 we may assume that α2,1= 1 . Using the Jacobi identity we have for all j≥2 0 = [e1,[e2, ej]] = −[e2,[ej, e1]] −[ej,[e1, e2]] = −αj,1[e1, e2]+[e1, ej] This implies [e1, ej] = αj,1[e1, e2] and [e1, αj,1e2−ej] = 0 for all j≥2 . Let us dene vj=αj,1e2−ej . Note that every vj lies in the center of g , and that the derived subalgebra D(g) is 1 -dimensional, generated by [e1, e2] . Now dene I:= ⟨[e1, e2], v3, . . . , vn⟩. This is an abelian subalgebra of dimension n−1 which contains the derived subalgebra D(g) . Hence I is an abelian ideal of maximal dimension n−1 , and we have β(g) = n−1 . Remark 2.1. Here, we give an alternative proof for the fact that g is almost abelian. Let a be an abelian subalgebra of dimension n−1 . We choose a complement space ⟨x⟩ with a⊕⟨x⟩=g . If [a,⟨x⟩]⊂a , then a is an abelian ideal of codimension 1 and we are done. If [a,⟨x⟩]*a , it is non-zero and contained in ⟨x⟩ , hence it is veried that [a,⟨x⟩] = ⟨x⟩ . Then, every two linear independent vectors in the Lie algebra g generate a 2 -dimensional subalgebra. By applying Lemma 5.3 of [111], we can conclude that g is almost abelian. Proposition 2.6. Let g be a supersolvable Lie algebra and let a be an abelian subalgebra of g . Suppose that b=a+⟨e1⟩ is a subalgebra of g , and that there is an x∈g such that [x, b]⊆b , but [x, a]⊆ a . Then, D(b) is one dimensional and Z(b) has codimension at most one in a . Proof. Let {ei}k i=2 be a basis for a such that e1= [x, e2] . Let [x, ej] = ∑k i=1 αj,iei for 1≤j≤k . Then [e2,[x, ej]] = αj,1[e2, e1] , so, for 2≤j≤k , 0 = [x, [e2, ej]] = −[e2,[ej, x]] −[ej,[x, e2]] = αj,1[e2, e1]−[ej, e1]. Hence [e1, ej] = αj,1[e1, e2] . It follows that D(b) = ⟨[e1, e2]⟩ . Put vj=αj,1e2−ej for 3≤j≤k . Then {vi}k i=3 ∈Z(b)∩a . Abelian subalgebras of codimension 1 17 The above result deals with the case where an abelian subalgebra of maximal dimension has codimension one in an ideal of g . Corollary 2.4. Let g be a supersolvable Lie algebra and let a be an abelian subalgebra of maximal dimension in g . If a⊂b where b is an ideal of g and a has codimension 1 in b , then α(g) = β(g) . Proof. If a is an ideal of g then the result is clear, so suppose that it is not. With the same notation as Proposition 2.6 the hypotheses of that result are satised. Then {v3, . . . , vk} ⊂ Z(b) ; in fact, the maximality of a gives Z(b) = ⟨v3, . . . , vk⟩ . Let c/⟨v3, . . . , vk⟩ be a chief factor of g (minimal ideal of g/⟨v3, . . . , vk⟩ ) with c⊂b . Then c is an abelian ideal of g with the same dimension as a . The result follows. Next we consider the situation where g has a maximal subalgebra that is abelian: rst when g is any non-abelian Lie algebra and K is algebraically closed, and then when g is solvable but K is arbitrary. Proposition 2.7. Let g be a non-abelian Lie algebra of dimension n over an algebraically closed eld K of any characteristic. Then, g has a maximal subalgebra a which is abelian if and only if g has an abelian ideal of codimension one in g . So α(g) = β(g) = n−1 . Proof. Suppose rst that g has a maximal subalgebra a which is abelian. If a is an ideal of g the proof is nished. So suppose that a is self-idealising, in which case it is a Cartan subalgebra of g . Let g=a˙ +g1(a) be the Fitting decomposition of g relative to a . Then {(ad m)|g1(a):m∈a} is a set of simultaneously triangulable linear mappings. So, there exists 0=b∈g1(a) such that [m, b] = λ(m)b for every m∈a , where λ(m)∈K . Then we have that a+⟨b⟩ is a subalgebra of g strictly containing a , whence a˙ +⟨b⟩=g and D(g) = ⟨b⟩ . But now dim(g/Cg(D(g))) = 1 and Cg(D(g)) = D(g)˙ +(a∩Cg(D(g))) is an abelian ideal of codimension one in g . The converse is clear. The following is a generalisation of [103, Proposition 3.1] Proposition 2.8. Let g be a solvable Lie algebra. Then g has a maximal subalgebra a which is abelian if and only if either (i) g has an abelian ideal of codimension one in g ; or (ii) C3(g) = ϕ(g) = Z(g) , C2(g)/C3(g) is a chief factor of g , and g splits over C2(g) . 18 Chapter 2. Theoretical Study Proof. Suppose rst that g has a maximal subalgebra a which is abelian. If a is an ideal of g we are in case (i) . So suppose that a is self-idealising, in which case it is a Cartan subalgebra of g . There is a k≥1 such that Ck(g)*a but Ck+1(g)⊆a . Then g=a+Ck(g) , which implies that C2(g)⊆ Ck(g) . It follows that C3(g)⊆a . If h is a subalgebra of g denote by eh its image under the canonical homomorphism onto g/C3(g) . Then ea is a Cartan subalgebra of eg and eg has a Fitting decomposition eg=ea˙ +eg1 . Now eg1⊆ C2(eg) = ] C2(g) , which is abelian, so eg1 is an ideal of eg . Moreover, since ea is a maximal subalgebra of eg , eg1 is a minimal ideal of eg and ] C2(g) = eg1 . It follows that C2(g)/C3(g) is a chief factor of g . Clearly ϕ(eg) = 0 , whence ϕ(g)⊆ C3(g) . Also g=a+C2(g) , so letting b be a subspace of a such that a=b˙ +(a∩C2(g)) we see that b is a subalgebra of g and g splits over C2(g) . Next, [a,C3(g)] ⊆ C2(a) = 0 , so a⊆Cg(C3(g)) . Since a is a self-idealising maximal subalgebra of g and Cg(C3(g)) is an ideal of g , we have Cg(C3(g)) = g , whence C3(g) = Z(g) . Finally, this means that C2(g) is nilpotent, giving C3(g) = ϕ(C2(g)) ⊆ϕ(g) by [99, Lemma 4.1 and Section 5], whence ϕ(g) = C3(g) . Consider now the converse. If (i) holds the converse is clear. So suppose that (ii) holds. Then g=b˙ +C2(g) , where b is an abelian subalgebra of g . Put a=b+C3(g) , so a is clearly abelian. Let a⊆c⊆g . Then ea⊆ec⊆eg . But eg=ea˙ +] C2(g) and ] C2(g) is a minimal abelian ideal of eg . So ea=ec implies that ec=eg . It follows that a is a maximal subalgebra of g . 2.3 Abelian subalgebras of codimension 2 Let g be a complex Lie algebra of dimension n satisfying α(g) = n−2 . We will show that g must be solvable except for the cases sl2(C)⊕Cℓ , for ℓ≥0 . We use the convention that the Lie algebra sl2(C) is included in this family, for ℓ= 0 . Proposition 2.9. Let g be an n -dimensional complex Lie algebra with α(g) = n−2 , then either g is isomorphic to one of the Lie algebras sl2(C)⊕Cℓ , or g is a solvable Lie algebra. Proof. Let g=snr be a Levi decomposition, where r denotes the solvable radical of g . For a semisimple Levi subalgebra s we have α(s)≤dim(s)−2, where equality holds if and only if s is sl2(C) . This follows from Table 1 and Lemma 2.1. By Lemma 2.3, we have that α(snr)≤α(s) + α(r) . Assume that s= 0 . Then Abelian subalgebras of codimension 2 19 it follows that α(g)≤α(s) + α(r)≤dim(s)−2 + dim(r) = n−2. Since we must have an equality, it follows that s is isomorphic to sl2(C) , and α(r) = dim(r) . Therefore r is abelian and g≃sl2(C)nϕCℓ with a homomorphism ϕ:sl2(C)→Der(Cℓ) . This Lie algebra contains an abelian subalgebra of codimension 2 if and only if ϕ is trivial. Indeed, the Lie bracket is given by [(x, a),(y, b)] = ([x, y], ϕ(x)b−ϕ(y)a) , for x, y ∈sl2(C) and a, b ∈Cℓ . Since there is an abelian subalgebra of codimension 2 , there must be a non-zero element (x, 0) commuting with all elements (0, b) , i.e., (0,0) = [(x, 0),(0, b)] = (0, ϕ(x)b) for all b∈Cℓ . It follows that ker(ϕ) is non-trivial. Since sl2(C) is simple, ϕ= 0 . In the other remaining case we have s= 0 . In that case, g is solvable. It is easy to classify such Lie algebras in low dimensions. Proposition 2.10. Let g be a complex Lie algebra of dimension n and α(g) = n−2 . (1) For n= 3 , it follows g≃sl2(C) . (2) For n= 4 , g is isomorphic to one of the following Lie algebras Table 2.1: Lie algebras of dimension 4 whose α invariant is 2 . g Lie brackets g1=r2(C)⊕r2(C) [e1, e2] = e2,[e3, e4] = e4 g2=sl2(C)⊕C[e1, e2] = e2,[e1, e3] = −e3,[e2, e3] = e1 g3[e1, e2] = e2,[e1, e3] = e3,[e1, e4] = 2e4,[e2, e3] = e4 g4(α), α ∈C[e1, e2] = e2,[e1, e3] = e2+αe3,[e1, e4] = (α+ 1)e4,[e2, e3] = e4 Proof. The proof is straightforward, using a classication of low-dimensional Lie algebras (see [19] for example). Note that g4(α)≃g4(β) if and only if αβ = 1 or α=β . Moreover, we can characterise solvable Lie algebras g whose biggest abelian subalgebras have codimension two in g . 20 Chapter 2. Theoretical Study Theorem 2.1. Let g be a solvable Lie algebra of dimension n with α(g) = n−2 , and let a be an abelian subalgebra of dimension n−2 . Then one of the following occurs (i) β(g) = n−2 ; (ii) g=C2(g)˙ +b , where b is an abelian subalgebra of g , C2(g) is the three-dimensional Heisenberg algebra, C3(g) = ϕ(g) = Z(g) and C2(g)/Z(g) is a two-dimensional chief factor of g (in which case β(g)≤n−3 ); (iii) a has codimension one in the nilradical, N , of g , which itself has codimension one in g . Moreover, C2(N) is one dimensional, Z(N) is an abelian ideal of maximal dimension and β(g) = n−3 . Proof. Let a be a maximal abelian subalgebra of g of dimension n−2 and suppose that (i) does not hold. Suppose rst that a is a maximal subalgebra of g . Then g is as in Proposition 2.8(ii) and a is a Cartan subalgebra of g . Let g=a˙ +g1 be the Fitting decomposition of g relative to a . Then g1⊆ C2(g) and dim g1= 2 . Let g1=⟨x, y⟩ . If [x, y] = 0 then g1 is an ideal of g and g/g1 is abelian, so C2(g)⊆g1⊆ C2(g) . This yields that g is metabelian and C2(g) is a two dimensional minimal ideal over which g splits. It follows from Proposition 2.3 that β(g)≥n−2 , a contradiction. If [x, y]= 0 , then C2(g) = ⟨[x, y]⟩+g1 and ⟨[x, y]⟩⊆C3(g) = Z(g) , so ⟨[x, y]⟩= Z(g) and we have case (ii). Let c be a maximal abelian ideal of g . Then Z(g)⊆c (Lemma 2.4) and C2(g)⊆ c . It follows that c∩ C2(g) = Z(g) . If dim c=n−2 , then dim(C2(g) + c) = dim C2(g) + dim c−dim(c∩C2(g)) = 3 + n−2−1 = n , so g=C2(g) + c . But then C2(g) = Z(g) , a contradiction. Hence, β(g)≤n−3 . So suppose that a is not a maximal subalgebra of g . Then a⊂d⊂g , where dim d=n−1 . Moreover, there is such a subalgebra a of d which is an ideal of d , by Proposition 2.5. Suppose rst that a does not act nilpotently on g . Then the Fitting decomposition of g relative to a is g=d˙ +g1 , and g1 is a one-dimensional ideal of g . Put b=a˙ +g1 , which is an ideal of g . Then Cb(g1) has codimension one in b and so is an abelian ideal of codimension two in g . It follows that β(g) = n−2 , a contradiction. Finally, suppose that a is an ideal of d and that a acts nilpotently on g . Then there is a k≥0 such that g( ad a)k⊆ d but g( ad a)k+1 ⊆d . Let x∈g( ad a)k\d , so g=d˙ +⟨x⟩ . Suppose rst that d is not an ideal of g . Then the core of d , dg has codimension one in d , by [1, Theorem 3.1]. If a=dg then we have case (i), so suppose that a=dg and d=a+dg . Then [a, x]⊆d which implies that [g,a]⊆d and [g,d] = [g,dg]+[g,a]⊆d ; that is, d is an ideal of g . Abelian subalgebras of codimension 2 21 Let N be the nilradical of g . If N⊆a then a⊆Cg(N)⊆N , so N=a and we have case (i) again. If a⊂N then N=g or we can assume that N=d . If a⊆ N and N⊆ a then either a+N=g , in which case g is nilpotent, or we can assume that a+N=d , in which case d is a nilpotent ideal of g and so d=N . If g is nilpotent, then we have case (i), as it will be seen in Proposition 2.11. So suppose that g=N+⟨e1⟩ , N=a+⟨e2⟩ and a=⟨e3, . . . , en⟩ . If a is an ideal of g then we have case (i) again, so suppose further that [e1, e3] = e2 and [e1, ej] = ∑n i=2 αj,iei for j≥2 . Following the proof of Proposition 2.6, we get [e2, ej] = αj,2[e2, e3] for j≥3 and C2(N) = ⟨[e2, e3]⟩ . Let [e2, e3] = ∑n i=2 λiei . Then [[e2, e3], e3] = λ2[e2, e3] , and nilpotency of N implies that λ2= 0 . It follows that [e2, e3]∈a . Similarly, [e2,[e2, e3]] = µ[e2, e3] (where µ=λ3+∑n j=4 λjαj,2 ), whence µ= 0 . We thus have that C3(N)=0 . Put vj=αj,2e3−ej for j≥3 . Then ⟨v3, . . . , vn⟩ ⊆ Z(N) , which is an abelian ideal of g . This is case (iii). Corollary 2.5. Let g be a supersolvable Lie algebra with α(g) = n−2 . Then β(g) = n−2 . Proof. We use the same notation as in Theorem 2.1 and show that cases (ii) and (iii) cannot occur. Clearly case (ii) cannot occur, since in that case C2(g)/Z(g) is a two-dimensional minimal ideal of g/Z(g) . So suppose that case (iii) occurs. Then dim Z(N) = n−3 . Since g is supersolvable, there is an ideal b⊂N of g with dim(b/Z(N)) = 1 . But clearly b is abelian, contradicting the maximality of Z(N) . Note that algebras of the type described in Theorem 2.1 (ii) and (iii) do exist over the real eld, as the following examples show. Example 2.3. Let g be the four-dimensional Lie algebra over R with basis {e1, e2, e3, e4} and non-zero products [e1, e2] = e3,[e1, e3] = −e2,[e2, e3] = e4. Then this algebra is as described in Theorem 2.1(ii). For C2(g) = ⟨e2, e3, e4⟩ is the three-dimensional Heisenberg algebra, g=C2(g)˙ +⟨e1⟩ , C3(g) = ⟨e4⟩=Z(g) = ϕ(g) , and C2(g)/Z(g) is a two-dimensional chief factor of g . This algebra has α(g) = 2 and β(g) = 1 . We could take a=⟨e1, e4⟩ , for example, but ⟨e4⟩ is the unique maximal abelian ideal of g . Example 2.4. Let g be the four-dimensional Lie algebra over R with basis {e1, e2, e3, e4} and non-zero products [e1, e2] = e2−e3,[e1, e4] = 2e4,[e1, e3] = e2+e3,[e2, e3] = e4. 28 Chapter 2. Theoretical Study Chapter 3 Abelian ideals and subalgebras in some solvable Lie algebras In this chapter, we show several results and algorithmic methods to compute abelian ideals and subalgebras of some special families of solvable Lie algebras over the eld K=C . More concretely, we deal with Lie algebras gn , given by n×n strictly upper-triangular matrices; Lie algebras hn , given by n×n upper-triangular matrices, Heisenberg algebras and liform Lie algebras. The content of this chapter can be seen in the papers [23] and [26]. The main interest to deal with the Lie algebras hn and gn lies in the fact that every nite-dimensional solvable Lie algebra can be represented by a Lie subalgebra of some hn (see [43, Theorem 9.11] or [105, Theorem 3.7.3]) and every nite-dimensional nilpotent Lie algebra is isomorphic to a subalgebra of gn ([105, Proposition 3.6.6]). Another reason is that their applications to Physic are many and varied as can be seen in [54, 87]. Heisenberg algebras constitute a special subclass of nilpotent Lie algebras and are also very interesting for several reasons. For example, their applications to both the theory of nilpotent Lie algebras itself and Theoretical Physics. With respect to the rst subject, a well-known result sets that a nilpotent Lie algebra g is abelian if and only if g does not contain a subalgebra isomorphic to the 3-dimensional Heisenberg algebra H1 [90]. Heisenberg algebras appeared at the beginnings of the 20th century when introducing Quantum Mechanics. In Classic Mechanics, the state of a particle in a given time t is determined by both its position vector Q∈R3 and its momentum vector P∈R3 . Heisenberg [52] took the components of these two vectors, considering them as operators in a Hilbert space such that the following 29 30 Chapter 3. Abelian ideals and subalgebras in some solvable Lie algebras commutation relations were veried [Qi, Qj] = 0,[Pi, Pj] = 0,[Pi, Qj] = −i~δij,∀i, j = 1,2,3, where δij is the named Kronecker delta and ~ , called the deformation parameter , represents the choice of a measure unit (usually ~= 1 ). Filiform Lie algebras constitute a very special subclass of nilpotent Lie algebras. In fact, they are the most structured Lie algebras in the nilpotent class and were introduced by Vergne [107] in 1966. The classication of solvable and nilpotent Lie algebras is still an open problem. Therefore, it seems convenient to reduce this problem by dealing with subclasses of nilpotent Lie algebras. Hence, studying and classifying liform Lie algebras is a rst step towards the classication of solvable Lie algebras in general. The method to compute an abelian subalgebra of maximal dimension of a Lie algebra g is based on the technique introduced in [6, 8] for subalgebras and consisting in considering an arbitrary basis of an r -dimensional subalgebra with respect to the basis Bg of the Lie algebra g in which its law is expressed. Then the vectors in Bg are divided in main vectors or non-main vectors according to a echelon matrix associated with Bg . Finally it is studied whether the subalgebra is abelian or not and if it is an ideal. 3.1 Lie algebra gn Now, we show an algorithmic procedure which allows us to obtain abelian ideals of the Lie algebra gn . A rst version of this algorithm for subalgebras was given in [6]. Case 1: n is even and n≥4 (i.e., n= 2 k , with k∈N\{1}) . Step 1: (2k)th column. Let us consider the 2k−1 vectors corresponding to this column. So, we obtain the abelian ideal ⟨X1,2k, . . . , X2k−1,2k⟩ . Step 2: (2k−1)th column. Now, we add the 2k−2 vectors corresponding to this column and the unique vector coming from the (2k−1)th row has to be removed. In this way, we obtain the abelian ideal ⟨X1,2k, . . . , X2k−2,2k, X1,2k−1, . . . , X2k−2,2k−1⟩. Step 2k−i+ 1 : ith column, with 2k > i > k+1 . After adding the i−1 vectors corresponding to the ith column, the 2k−i vectors corresponding to ith row are removed. In this way, the dimension of the obtained abelian ideal Lie algebra gn 31 increases 2i−2k−1 , while i > k +1/2 . So, k is the last step in which the number of vectors generating the obtained abelian ideal is greater than the one obtained in the previous step. Step k : (k+ 1)th column. The k vectors corresponding to this column are added, whereas the 2k−(k+ 1) = k−1 ones corresponding to the (k+1)th row are removed. So, a k2 -dimensional abelian ideal is obtained, being its basis formed by X1,k+1 . . . X1,2k X2,k+1 . . . X2,2k . . ..... . . Xk,k+1 . . . Xk,2k Case 2: n is odd and n≥4 (i.e., n= 2k+ 1 , with k∈N\{1}) . Analogously to the Case 1, we can settle the following procedure to obtain an abelian ideal with dimension as large as possible. Step 1: (2k+ 1)th column. Firstly, we consider the 2k vectors corresponding to this column. So we obtain the abelian ideal ⟨X1,2k+1, . . . X2k,2k+1⟩ . Step 2 : 2kth column. Now the 2k−1 vectors corresponding to this column are added, removing the vector coming from the 2kth row. In this way, the abelian ideal ⟨X1,2k+1, . . . X2k−1,2k+1, X1,2k, . . . X2k−1,2k⟩ is obtained. Step 2k−i+ 2 : ith column, with 2k+1 > i > k+2 . There exist i−1 vectors corresponding to the ith column, which are added to the generators of the abelian ideal obtained in the previous step. Then the 2k+ 1 −i vectors corresponding to the ith row are removed, obtaining an abelian ideal whose dimension increases 2i−2k−2 with respect to the previous step. The dimension increases only when i > k + 1 . Hence, Step k is the last step and the procedure will be stopped after it. Step k : (k+2)th column. Now the k+1 vectors corresponding to the (k+2)th column are added, whereas the k−1 vectors corresponding to the (k+2)th row are removed, which allows us to obtain a (k2+k) -dimensional abelian ideal with the basis X1,k+1 . . . X1,2k+1 X2,k+1 . . . X2,2k+1 . . ..... . . Xk,k+1 . . . Xk,2k+1 32 Chapter 3. Abelian ideals and subalgebras in some solvable Lie algebras By using a similar procedure we can compute this other (k2+k) -dimensional abelian ideal X1,k+2 . . . X1,2k+1 X2,k+2 . . . X2,2k+1 . . ..... . . Xk+1,k+2 . . . Xk+1,2k+1 According to the results obtained, an abelian ideal of gn has been computed for all n∈N . Indeed, the dimension of such a ideal is An=     n−1, if n < 4; k2, if n= 2k, n ≥4; k2+k, if n= 2 k+ 1, n ≥4. Therefore, α(gn) = β(gn) is lower bounded by the value An . Now we prove that the value of the invariants α and β for gn is An , improving the bound given by Jacobson [57] and Schur [91]. Let us remember that dgn denotes the dimension of gn . Now, we give a sketch for the proof of the following fact: it is not possible to obtain an abelian subalgebra of the Lie algebra gn with dimension greater than An . A more detailed information about this proof can be consulted in [8]. For it, three preliminary results are needed Proposition 3.1 (Theorem 3.1 in [6]) . Let us consider n∈N , with n≥4 . Then the abelian Lie algebra of dimension dgn−1 is not a subalgebra of gn .  Lemma 3.1 (Lemma 4.1 in [8]) . Let us consider n∈N , with n≥3 . Then, dgn−dgn−1=n−1 .  Lemma 3.2 (Lemma 4.2 in [8]) . Let us consider n∈N , with n≥3 . Then, An−An−1=[n 2] , where [n 2] denotes the integral part of n 2 .  Combining the previous results, a proof can be provided for Theorem 3.1 (Theorem 4.1 in [8]) . Let us consider n∈N , with n≥2 . The α invariant of the Lie algebra gn is given by α(gn) = An={k2,if n= 2k, with k∈N, k2+k, if n= 2k+ 1,with k∈N. Proof. We use an iterative procedure based on studying the impossibility of obtaining a ( dgn−r )-dimensional abelian Lie subalgebra in gn for 0< r ≤dgn−An−1 . Cases g2 and g3 are trivial. Moreover, Proposition 3.1 implies this result for g4 . So we only need to prove it for n≥5 . Given an abelian subalgebra of gn , our proof is based on the number of main vectors coming from the nth column in an arbitrary basis B of that subalgebra. Lie algebra gn 33 ( dgn−2 )-dimensional abelian subalgebra: We have to consider three dierent cases Case 1: The two non-main vectors correspond to the nth column. Each vector Xi,j ∈gn−1 is main, but gn−1 is not abelian. Case 2: A unique non-main vector corresponds to the nth column. A (dgn−1− 1) -dimensional abelian subalgebra has to be contained in gn−1 , but it is not possible according to Proposition 3.1. Case 3: No non-main vector corresponds to the nth column. [Xi0,n−1, Xn−1,n] is non-zero where Xi0,n−1 is a main vector, with i0 as greater as possible. ( dgn−3 )-dimensional abelian subalgebra: There are three non-main vectors in B and the four following cases have to be considered Case 1: The three non-main vectors correspond to the nth column. This is in contradiction with the law of gn−1 . Case 2: Two non-main vectors correspond to the nth column. Due to Proposition 3.1, we cannot nd a ( dgn−1−1 )-dimensional abelian subalgebra in gn−1 . Case 3: A unique non-main vector corresponds to the nth column. Only one vector coming form the nth column of gn is non-main. So we need to nd a ( dgn−1−2 )-dimensional abelian subalgebra in gn−1 . • If Xn−1,n is main, [Xi0,n−1, Xn−1,n] is non-zero where Xi0,n−1 is a main vector with i0 as greater as possible. • If Xn−1,n is non-main, Xi,n is main for 1≤i≤n−2 . If n= 5 , we choose the main vectors X2,5 and X1,2 to obtain a non-zero bracket. If n > 5 , [Xj0,n−2, Xn−2,n] is non-zero with j0 as greater as possible such that Xj0,n−2 is a main vector. Case 4: No non-main vector corresponds to the nth column. If n > 5 , there exists i0 as greater as possible such that Xi0,n−1 is main and the bracket [Xi0,n−1, Xn−1,n] is non-zero. If n= 5 , we consider the main vectors X3,5 and X1,3 to get a non-zero bracket. 34 Chapter 3. Abelian ideals and subalgebras in some solvable Lie algebras ( dgn−r )-dimensional abelian subalgebra, where 4≤r≤dgn−An−1 : We have to prove the non-existence of ( dgn−r )-dimensional abelian subalgebras in gn , with 4≤r≤dgn−An−1 . Our proof is based on an iterative procedure supposing that, for 1≤s≤r−1 , no abelian subalgebra of dimension dgn−s can be contained in gn . Remember that s≤dgn−An . We consider ve cases Case 1: r non-main vectors correspond to the nth column. Then the algebra gn−1 would be abelian and this is not possible. Case 2: r−1 non-main vectors correspond to the nth column. There exists one non-main vector in gn−1 and a (dgn−1−1) -dimensional abelian subalgebra is contained in gn−1 . It is not possible either due to Proposition 3.1. Case 3: r−k non-main vectors correspond to the nth column (with 2≤k≤ min{r−1, dgn−1−An−1−1} ). There are k non-main vectors in gn−1 and a ( dgn−1−k )-dimensional abelian subalgebra is looked for in gn−1 . This stands in contradiction with our iterative assumption. Case 4: r−k non-main vectors correspond to the nth column (with dgn−1− An−1≤k≤r−1 ). There exist k non-main vectors in gn−1 and n−1−r+k main ones coming from the nth column, satisfying 1+[n 2]≤n−1−r+k≤n−1 . Thus, there are, at least, 1+[n 2] main vectors with the form Xi,n and, at most, n−[n 2]−2 non-main ones. So, more than the half of the vectors coming from the nth column are main. Now, we choose w∈ B such that its main vector Xj0,h0 , with 1≤j0< h0< n , belongs to gn−1 and the vector Xh0,n is the main vector of a basis vector v∈ B . So the bracket [v,w] is non-zero. To assert it, j0 has to be chosen as greater as possible. Case 5: No non-main vector corresponds to the nth column. There exists a main vector Xi0,j0∈gn−1 with i0 as greater as possible and [Xi0,j0,Xj0,n]=0 .  3.2 Lie algebra hn We explain an algorithmic method to obtain abelian ideals in an arbitrary Lie algebra hn , with n≥4 . The content of this section can be seen in [21]. Depending on the parity of n , two possible cases have to be considered Lie algebra hn 35 Case 1: n is even and n≥4 (i.e., n= 2 k , with k∈N\{1}) . We consider the vectors in the basis of hn , corresponding to the columns in the matrix expression of hn . When the vectors corresponding to the ith column are chosen, all the vectors corresponding to the ith row have to be removed. In this way, all the non-zero brackets are avoided and it is an ideal. Step 1: (2k)th column. We add the 2k vectors corresponding to the (2k)th column. The vector coming from the (2k)th row has to be removed, obtaining the abelian subalgebra ⟨X1,2k, . . . , X2k−1,2k⟩ . Step 2k−i+ 1 : ith column, with 2k > i > k + 1 . After adding the i vectors corresponding to the ith column, the 2k−(i−1) vectors corresponding to ith row are removed. In this way, we obtain an abelian ideal whose dimension increases 2i−2k−1 , while i > k + 1/2 . So k is the last step in which the dimension increases with this adding-removing procedure. Step k : (k+ 1)th column. The k+ 1 vectors corresponding to the (k+ 1)th column are added, whereas the k ones corresponding to the (k+ 1)th row are removed. Step k+1 : Adding the vector ∑n i=1Xi,i to the basis computed in Step k , we obtain a (k2+ 1) -dimensional abelian ideal whose basis is shown next X1,k+1 . . . X1,2k X2,k+1 . . . X2,2k . . ..... . . Xk,k+1 . . . Xk,2k and n ∑ i=1 Xi,i Case 2: n is odd and n≥4 (i.e., n= 2k+ 1 , with k∈N\{1}) . By arguing analogously to the Case 1, we can settle the following procedure to obtain an abelian ideal with dimension as large as possible. Step 1: (2k+ 1)th column. After adding the 2k+ 1 vectors corresponding to this column, the unique vector coming from the (2k+1)th row is removed, obtaining the abelian ideal ⟨X1,2k+1, . . . , X2k,2k+1⟩ . Step 2k−i+ 2 : ith column, with 2k+1 > i > k+2 . The i vectors corresponding to the ith column are added and the 2k−(i−1) vectors corresponding 36 Chapter 3. Abelian subalgebras in some solvable Lie algebras to the ith row are removed, obtaining an abelian ideal whose dimension increases 2i−2k−2 . This dimension increases while i>k+ 1 . Hence, Step k is the last step for the adding-removing procedure. Step k : (k+2)th column. Now the k+2 vectors corresponding to this column are added and, then, the 2k+ 1 −(k+ 2 −1) = k vectors corresponding to the (k+ 2)th row are removed, which allows us to obtain a (k2+k) - dimensional abelian ideal. Step k+ 1 : Adding the vector ∑n i=1 Xi,i to the basis computed in Step k , we obtain the (k2+k+ 1) -dimensional abelian ideal determined by the basis X1,k+2 . . . X1,2k+1 X2,k+2 . . . X2,2k+1 . . ..... . . Xk+1,k+2 . . . Xk+1,2k+1 and n ∑ i=1 Xi,i By using a similar procedure we can compute this other (k2+k+1) -dimensional abelian ideal X1,k+1 . . . X1,2k+1 X2,k+1 . . . X2,2k+1 . . ..... . . Xk,k+1 . . . Xk,2k+1 and n ∑ i=1 Xi,i According to the results obtained in this section, an abelian ideal of hn has been computed for all n∈N . Indeed, the dimension of such a subalgebra is Bn=   n, if n < 4; k2+ 1, if n= 2k, n ≥4; k2+k+ 1, if n= 2 k+ 1, n ≥4. Therefore, we can arm that α(hn) = β(hn) is lower bounded by the value Bn , which is equal to the upper bound of α(hn) given by Jacobson [57] and Schur [91]. So, for the Lie algebras hn , α and β invariants are equal to Bn . 3.3 Heisenberg algebras This subsection is devoted to compute the α invariant for complex Heisenberg algebras. To do so, lower and upper bounds of α(Hk) are computed for k∈N using the Heisenberg algebras 37 proof introduced in [78]. According to the law of Hk and the basis Bk={Xi}2k+1 i=1 expressed in preliminaries, ak=⟨{X2i+1|i= 0, . . . , k}⟩ is a (k+ 1) -dimensional abelian ideal of Hk . So we can arm that k+ 1 ≤α(Hk) for k∈N . In this way, our main goal comes down to proving the non-existence of abelian subalgebras in Hk with dimension greater than k+ 1 . The following technical lemma has to be used to settle the previous statement, whose proof is based in solving a system with a symbolic computation package Lemma 3.3. The system of equations with complex unknowns ai and bi has not any complex solutions E1:a1·b2−a2·b1=−1; E6:a2·b4−a4·b2= 0; E2:a1·b3−a3·b1= 0; E7:a2·b5−a5·b2= 0; E3:a1·b4−a4·b1= 0; E8:a3·b4−a4·b3=−1; E4:a1·b5−a5·b1= 0; E9:a3·b5−a5·b3= 0; E5:a2·b3−a3·b2= 0; E10 :a4·b5−a5·b4= 0. (3.1)  By using the previous lemma, the following theorem can be proved, settling lower and upper bounds for α(Hk) = β(Hk) . Theorem 3.2 (Theorem 3.1 in [78]) . For k∈N and γ∈N∪{0} , α(Hk) = β(Hk) is bounded by k+ 1 ≤α(Hk)≤2k−γ , for all k≥γ+ 1 . Proof. The rst inequality is trivial due to the existence of the subalgebra ak for k∈N . To obtain the upper bound for α(Hk) , we only need to prove the non-existence of abelian subalgebras b of Hk whose dimension is 2k−γ+1 . Since k is lower bounded by γ+ 1 , the dimension of Hk can be lower bounded by dim(Hk) = 2k+ 1 ≥2γ+ 3. (3.2) Any ( 2k−γ+ 1 )-dimensional subalgebra of Hk has ( 2k−γ+ 1 ) main vectors and the number of the non-main ones is dim(Hk)−(2k−γ+ 1) = γ . By combining this expression with (3.2), the number of main vectors is greater than the number of non-main ones; that is 2k−γ+ 1 ≥γ+ 3 . The basis Bk of Hk is structured in pairs of basic vectors (Xi, Xi+1) such that i is an even natural number less than or equal to 2k . For each of these pairs, its corresponding bracket is equal to the vector X1∈ Bk . Let us call non-zero pair to each of these pairs. Therefore, Hk is formed by k non-zero pairs and the vector resulting from all these brackets. There are, at most, ⌊γ 2⌋ non-zero pairs which are formed by non-main vectors; whereas the number of non-zero pairs which only contain main vectors is at least 44 Chapter 3. Theoretical Study Since (z2−2z1+2 0)=(z2−z1 0)= 1 , the coecient of en−2z1+3 is α1·(z2−2z1+2 ∑ k=1 (z2−2z1+2 k)Pn−z2+1([ez1+k, en−k]) − z2−z1 ∑ k=1 (z2−z1 k)Pn−z2+1([ez1+k, en−k])). Let us note that z2−z1> z2−2z1+ 2 , since z1>2 . Consequently, the coecient of en−2z1+3 is α1·(z2−2z1+2 ∑ k=1 ((z2−2z1+2 k)−(z2−z1 k))Pn−z2+1([ez1+k, en−k]) − z2−z1 ∑ k=z2−2z1+3(z2−z1 k)Pn−z2+1([ez1+k, en−k])). Now, we consider the family of functions {Φh:Z→N∪{0}}h∈N∪{−1,0} , dened as Φ−1(n) = {1,if n∈N; 0,otherwise.and Φh(n) =      n ∑ i=1 Φh−1(i),if n∈N; 0,otherwise.∀h∈N∪{0}. With this notation, we obtain Pn−z2+1([ez1+k, en−k]) = (k ∑ l=0 (k l)Φk−1−l(n−z2−2k+l))α1. and the coecient of en−2z1+3 turns out to be (α1)2·(z2−2z1+2 ∑ k=1 ((z2−2z1+2 k)−(z2−z1 k)) k ∑ l=0 (k l)·Φk−1−l(n−z2−2k+l) − z2−z1 ∑ k=z2−2z1+3(z2−z1 k)k ∑ l=0 (k l)Φk−1−l(n−z2−2k+l)). If n−z2−2k+l≤0 for 1≤k≤z2−z1 and 0≤l≤k , then the Jacobi identity J(ez2, ez2+1, en)=0 is identically null. Otherwise, the previous expression is zero if and only if α1= 0 . Next, we have this result for the case β(g) = n−3 . Proposition 3.8. Let g be a n -dimensional liform Lie algebra verifying β(g) = n−3 . Then the triples ([n−(2k+1) 2]+2, n −2, n) , where n≥2k+ 5 , correspond to a liform Lie algebra with unique restrictions given by {αi= 0}k i=1 . Filiform Lie algebras 45 Proof. In this case, we have to impose the Jacobi identity J(en−2, en−1, en) = 0 over the general law given in Proposition 3.3 with (z1, z2, n) = ([n−(2k+1) 2] + 2, n −2, n) . This identity is given by [[en−2, en−1], en] + [[en, en−2], en−1]=0 . Next, we compute both brackets, starting with [[en−2, en−1], en] = [α1en−z1+α2en−z1−1+. . . +αn−2−z1e3+αn−z1−1e2, en] = [α1en−z1+α2en−z1−1+. . . +αn−2z1+1ez1, en] =α1[en−z1, en] + α2[en−z1−1, en] + . . . +α2k−2[ez1, en] =α1((2k−2)α1e2k+ ((2k−3)α2+α1 2)e2k−1 +((2k−4)α3+α2 3)e2k−2+. . .+(α2k−2+α2k−3 2k−2)e3+α2k−2 2k−1e2) +α2((2k−3)α1e2k−1+ ((2k−4)α2+α1 2)e2k−2 +((2k−5)α3+α2 3)e2k−3+. . .+(α2k−3+α2k−4 2k−3)e3+α2k−3 2k−2e2) +. . . +α2k−2(α1e3+α1 2e2). and continuing with the second taking into account that [[en, en−2], en−1] = −[[en−2, en], en−1] [[en−2, en], en−1] = [(n−z1−1)α1en−z1+1 + ((n−z1−2)α2+α1 2)en−z1+... +(αn−z1−1+αn−z1−2 n−z1−1)e3+αn−z1−1 n−z1e2, en−1] = (n−z1−1)α1[en−z1+1, en−1]+((n−z1−2)α2+α1 2)[en−z1, en−1] +. . . + (αn−z1−1+αn−z1−2 n−z1−1)[e3, en−1] + αn−z1−1 n−z1[e2, en−1] = (n−z1−1)α1[en−z1+1, en−1]+((n−z1−2)α2+α1 2)[en−z1, en−1] +. . . + ((z1−2)α2k−1+α2k−2 2k−1)[ez1, en−1] = (n−z1−1)α1(α1e2k+α2e2k−1+. . . +α2k−1e2) +((n−z1−2)α2+α1 2)(α1e2k−1+α2e2k−2 +. . . +α2k−2e2) + . . . + ((z1−2)α2k−1+α2k−2 2k−1)α1e2. Combining both brackets, the Jacobi identity J(en−2, en−1, en) = 0 is α1((2k−2)α1e2k+ ((2k−3)α2+α1 2)e2k−1+ ((2k−4)α3+α2 3)e2k−2+. . . +(α2k−2+α2k−3 2k−2)e3+α2k−2 2k−1e2)+α2((2k−3)α1e2k−1+((2k−4)α2+α1 2)e2k−2 +((2k−5)α3+α2 3)e2k−3+. . . + (α2k−3+α2k−4 2k−3)e3+α2k−3 2k−2e2) + . . . +(α2k−3+α2k−4 2k−3)e3+α2k−3 2k−2e2) + . . . +α2k−2(α1e3+α1 2e2) −(n−z1−1)α1(α1e2k+α2e2k−1+. . . +α2k−1e2) −((n−z1−2)α2+α1 2)(α1e2k−1+α2e2k−2+. . . +α2k−2e2) −((n−z1−3)α3+α2 3)(α1e2k−2+α2e2k−3+. . . +α2k−3e2)−. . . −((z1−2)α2k−1+α2k−2 2k−1)(α1e2) = 0. Note that, according to the previous computations, the coecient of the vector e2k is (α1)2·(2k−n+z1−1) . Fixing 0< h ≤2k−2 , we compute the coecient of 46 Chapter 3. Theoretical Study the vector e2k−h as follows αh+1(2k−(h+2))α1+ h ∑ i=1 αi((2k−(h+2))αh+2−i+αh+1−i h+2−i)−(n−z1−1)α1αh+1 − h+1 ∑ i=2 ((n−z1−i)αi+αi−1 i)αh+2−i=(2k−(h+2))α1αh+1+ h ∑ i=1 αiαh+2−i(2k−(h+2)) −(n−z1−1)α1αh+1 − h+1 ∑ i=2 αiαh+2−i(n−z1−i) = (2k−n−h+ 1)α1αh+1 + h ∑ i=2 αiαh+2−i(2k−n−h−2 + z1+i). In this way, we have to consider the following equations for 0< h ≤2k−2 ,      (α1)2(2k−n+z1−1) = 0, (2k−n−h+ 1)α1αh+1 + h ∑ i=2 αiαh+2−i(2k−n−h−2 + z1+i) = 0. From the rst equation, we obtain α1= 0 , since n≥2k+ 5 . Consequently, the second equation turns into h ∑ i=2 αiαh+2−i(2k−n−h−2 + z1+i) = 0,∀2< h ≤2k−2. This involves the condition αl= 0 for h= 2l−2 , where l= 2,3, . . . , k , leading to the conclusion that the Jacobi identity J(en−2, en−1, en) = 0 implies {αi= 0}k i=1 . Moreover, we can set the following result for the coecients in a liform Lie algebra associated with a general triple (z1, z2, n) Proposition 3.9. Let g be a n -dimensional non-model liform Lie algebra associated with the triple (z1, z2, n) . All the Jacobi identities in g correspond to a system of equations whose solutions are a combination of the following expressions (βαp+γαp−1 p)(λαq+µαq−1 q) = 0,where β, γ, λ, µ ∈Cand p, q ∈ {2, . . . , n −z2} Therefore, we always obtain linear relations between the coecients in the law of g . Proof. Let g be a liform Lie algebra associated with (z1, z2, n) . We suppose that there are restrictions over the coecients {α1, α2, . . . , αz2−z1+1, α1 2, . . . , αn−z2−1 n−z2} . By using Proposition 3.7, we can take α1= 0 . The general law of g is given by Filiform Lie algebras 47 [e1, eh] = eh−1,∀3≤h≤n [ez1+i, ez2+1] = α2ei+1 +. . . +αi+1e2,1< i ≤z2−z1. (3.6) [ez1, ez2+j] = α1 2ej. . . +αj−1 je2,2≤j≤n−z2. (3.7) [ez1+p, ez2+k] = p+k ∑ h=2 Ph([ez1+p−1, ez2+k]+[ez1+p, ez2+k−1])eh+1 +αk+p−1 k+pe2, 2≤k≤n−z2,0< p < z2−z1+k. (3.8) The family of brackets (3.6), (3.7) and (3.8) will be called brackets of type 1 , 2 and 3 , respectively. We need to impose the Jacobi identities J(ea, eb, ec) = 0 , where z1≤a < b < c ≤n . According to the law of g , we have to study the identities involving the brackets of type 1 , 2 and 3 . Therefore, we study the following Jacobi identities J(ez1+i, ez1+i′, ez2+1) = 0, J(ez1, ez1+i, ez2+1) = 0, J(ez1, ez1+i, ez2+j) = 0, J(ez1, ez2+1, ez2+j) = 0, J(ez1+i, ez2+1, ez2+j) = 0, J(ez1+i, ez2+j, ez2+k) = 0. • For J(ez1+i, ez1+i′, ez2+1) = 0 , all the brackets involved are of type 1 . This identity is given by [[ez1+i, ez1+i′], ez2+1] + [[ez1+i′, ez2+1], ez1+i] + [[ez2+1, ez1+i], ez1+i′]. The bracket [ez1+i, ez1+i′] = 0 since ⟨e2, . . . , ez2⟩ is an abelian ideal. Therefore, J(ez1+i, ez1+i′, ez2+1) = 0 is expressed as follows [α2ei′+1 +. . . +αi′+1e2, ez1+i]−[α2ei+1 +. . . +αi+1e2, ez1+i′]. and both brackets are zero since i′+ 1, i + 1 ≤z2−z1+ 1 ≤z2−3< z2+ 1 . Consequently, this Jacobi identity is identically null. • The Jacobi identity J(ez1, ez1+i, ez2+1) = 0 , with i≤z2−z1 , is given by [[ez1, ez1+i], ez2+1] + [[ez1+i, ez2+1], ez1] + [[ez2+1, ez1], ez1+i], where [ez1, ez2+1] = [ez1, ez1+i] = 0 since α1= 0 and ⟨e2, . . . , ez2⟩ is an abelian ideal. Therefore, the identity is given by [[ez1+i, ez2+1], ez1] = [α2ei+1 +. . . + 48 Chapter 3. Theoretical Study αi+1e2, ez1] . Let us note that if i+1 ≤z2 , then this bracket is null. We consider i+ 1 > z2 . In that case, we have [[ez1+i, ez2+1], ez1] = −[ez1, α2ei+1 +...+αi+2−z2ez2+1] = −αi+2−z2[ez1, ez2+1] −αi+1−z2[ez1, ez2+2]−. . . −α2[ez1, ei+1] =−αi+1−z2α1 2e2−αi−z2(α1 2e3+α2 3e2)−. . . −α2(α1 2ei+2−z2+. . . +αi+1−z2 i+2−z2e2) =−α2α1 2ei+2−z2−(α2α2 3+α3α1 2)ei+1−z2−. . . −(α2αi−z2−1 i−z2+. . . +αi+1−z2α1 2)e2= 0. obtaining the following system of equations                α2α1 2= 0, α2α2 3+α3α1 2= 0, α2α3 4+α3α2 3+α4α1 2= 0, . . .. . . α2αi−z2−1 i−z2+. . . +αi+1−z2α1 2= 0. which only returns solutions of type αp= 0 or αq−1 q= 0 . • For J(ez1, ez1+i, ez2+j) = 0 , we have the following expression [[ez1, ez1+i], ez2+j] + [[ez1+i, ez2+j], ez1] + [[ez2+j, ez1], ez1+i]. Let us note that [ez1, ez1+i] = 0 since ⟨e2, . . . , en⟩ is an abelian ideal. Moreover, [ez1, ez2+j] = α1 2ej+. . .+αj−1 je2 with 2≤j≤n−z2 . From j≤n−z2≤z2−2< z2 and the denition of z2 , we can arm that [[ez1, ez2+j], ez1+i] = 0 . Hence, the Jacobi identity J(ez1, ez1+i, ez2+j) = 0 is given by [[ez1+i, ez2+j], ez1] = 0 . Developing the rst term, we obtain [[ez1+i, ez2+j], ez1] = [∑i+j h=2 Ph([ez1+i−1, ez2+j]+[ez1+i, ez2+j−1])eh+1 +αi+j−1 i+je2, ez1] = [∑i+j h=z2+1 Ph(u)eh+1, ez1] = −Pz2+1(u)[ez1, ez2+2]−. . . −Pi+j(u)[ez1, ei+j+1] = −Pz2+1(u)(α1 2e2)−. . . −Pi+j(u)(α1 2ei+j+1−z2+...+αi+j−z2 i+j+1−z2e2) =−α1 2Pi+j(u)ei+j+1−z2−(α2 3Pi+j+α1 2Pi+j−1)(u)ei+j−z2−. . . −(αi+j−z2 i+j+1−z2Pi+j+. . . +α1 2Pz2+1)(u)e2= 0, where u= [ez1+i−1, ez2+j]+[ez1+i, ez2+j−1] . Consequently, we have to solve the system of equations            α1 2Pi+j(u) = 0, α2 3Pi+j(u) + α1 2Pi+j−1(u) = 0, . . .. . . αi+j−z2 i+j+1−z2Pi+j(u) + . . . +α1 2Pz2+1(u) = 0. Filiform Lie algebras 49 Now, we compute the expression of Pi+j(u), Pi+j−1(u), . . . , Pz2+1(u) by using the reasoning of the proof of Proposition 3.7. Pi+j(u) = α1 i ∑ k=0 k ∑ l=0 (i k)(k l)Φk−1−l(z2+j−2k+l); Pi+j−1(u) = α1 2 i ∑ k=0 k ∑ l=0 (i k)(k l)Φk−1−l(z2+j−2k+l); . . .. . . Pz2+1(u) = αi+j−z2−1 i+j−z2 i ∑ k=0 k ∑ l=0 (i k)(k l)Φk−1−l(z2+j−2k+l). When solving this system of equations, we obtain solutions of type αq−1 q= 0 . • For the Jacobi identities J(ez1, ez2+1, ez2+j) = 0, J(ez1+i, ez2+1, ez2+j) = 0 and J(ez1+i, ez2+j, ez2+k)=0 , only brackets of type 3 are involved. We will study this type of brackets. We dene the degree function deg : N×N→N , where N={1, . . . , n} , as deg(a, b) = a+b . Notice that, according to Proposition 3.3, deg([a, b]) = a+b implies [ea, eb]∈ ⟨e2, . . . , ea+b−z1−z2⟩ . The brackets of type 3 are given by [ez1+p, ez2+k] = p+k ∑ h=2 Ph([ez1+p−1, ez2+k] + [ez1+p, ez2+k−1])eh+1 +αk+p−1 k+pe2 Let us note that deg(z1+p−1, z2+k) = z1+z2+p+k−1 = deg(z1+p, z2+k−1) . In order to compute the term ∑p+k h=2 Ph([ez1+p−1, ez2+k]+[ez1+p, ez2+k−1])eh+1 in a bracket of type 3 , we have to decompose it by using the following brackets of type 1 and 2 [ez1+l, ez2+1] = α2el+1 +. . . +αl+1e2,1≤l≤z2−z1; [ez1, ez2+l+1] = α1 2el+1 +. . . +αl l+1e2,2≤l+ 1 ≤n−z2, and their index pairs have the same degree. In fact, we obtain that Ph([ez1+l, ez2+1]+[ez1, ez2+l+1])=(αl−h+3+αl−h+2 l−h+3),for {1≤l≤z2−z1; 2≤h≤l+1. Ph([ez1+l, ez2+1]+[ez1, ez2+l+1])=αl−h+2 l−h+3,for {z2−z1<l≤n−z2−1; 2≤h≤l+1. From the remaining Jacobi identities, we obtain this type of equations (βαp+γαp−1 p)(λαq+µαq−1 q)=0, β, γ, λ, µ∈C,and p, q∈{2, . . . , n −z2}. 50 Chapter 3. Theoretical Study Now, we show an algorithmic procedure which computes the law of an n -dimensional non-model liform Lie algebra g starting from the value of α(g) . More concretely, we give a step-by-step explanation of this algorithm for obtaining the law of g with α(g) = k∈N . Input 1. The dimension n of a non-model liform Lie algebra g . 2. The value k of the invariant α(g) . Output 1. A list with the triples (z1, z2, n) such that there exist non-model liform Lie algebras associated with them. 2. The law of each family of liform Lie algebras for each triple. Method 1. First, computing the value of the invariant z2 by using Proposition 3.2. According to this value and Expression (1.2), several possibilities appear for the invariant z1 . 2. For each value of z1 , computing all the possible non-zero brackets of g given by Proposition 3.3. 3. Then, ruling out those values of z1 not satisfying the Jacobi identities J(eh, ek, el) = 0 , for z1≤h < k < l ≤n . 4. Next, obtaining a list with all the triples (z1, z2, n) such that there exist nonmodel liform Lie algebras having such invariants. 5. By using Proposition 3.3 again, computing the law of each Lie algebra associated with a triple given in the previous step. Now, we show the implementation of the previous algorithm. To do so, we have used the symbolic computation package MAPLE 12 . Note that before running any sentence or loading a package, we must restart all the variables and delete all the computations with the command restart . After that, we load the library DifferentialGeometry, LieAlgebras to activate commands related to Lie algebras such as BracketOfSubspaces . Filiform Lie algebras 51 First, we show the implementation of a routine which computes the law of a liform Lie algebra from the triple (z1, z2, n) by using Proposition 3.3. This routine, named law , receives the triple (z1, z2, n) as input and returns the liform Lie algebra associated with this triple. For the implementation, we dene a list as a local variable L . This list saves the indexes and the value of the structure constants corresponding with the non-zero brackets of the liform Lie algebra. First, L saves the brackets given by liformity (i.e. [e1, eh] = eh−1,∀3≤h≤n ) and the bracket [ez1, ez2+1] = α1e2 . Then, a loop is programmed for including the indexes of the rest of non-zero brackets in L . > law:=proc(z_1,z_2,n) > local L; L:=[seq([[1,h,h-1],1],h=3..n), > [[z_1,z_2+1,2],a[1]]]; > for i from 1 to n do for j from 1 to n do c[i,i][j]:=0; od; od; > for i from 1 to z_2-z_1 do for l from 1 to i+1 do > L:=[op(L),[[z_1+i,z_2+1,i+3-l],a[l]]]; > c[z_1+i,z_2+1][i+3-l]:=a[l]; > od; od; > for i from 2 to n-z_2 do > L:=[op(L),[[z_1,z_2+i,i+1],a[1]]]; > c[z_1,z_2+i][i+1]:=a[1]; > for j from 2 to i do > L:=[op(L),[[z_1,z_2+i,i+2-j],a[j][j-1]]]; > c[z_1,z_2+i][i+2-j]:=a[j][j-1]; > od; od; > for i from 2 to n-z_2 do for j from 1 to z_2-z_1+i-1 do > L:=[op(L),[[z_1+j,z_2+i,2], > a[i+j][i+j-1]]]; > c[z_1+j,z_2+i][2]:=a[i+j][i+j-1]; > for h from 2 to i+j do > L:=[op(L),[[z_1+j,z_2+i,h+1], > c[z_1+j-1,z_2+i][h]+c[z_1+j,z_2+i-1][h]]]; > c[z_1+j,z_2+i][h+1]:=c[z_1+j-1,z_2+i][h]+ > c[z_1+j,z_2+i-1][h]; > od;od;od; > return _DG([["LieAlgebra",Alg1,[n]],L]); > end proc: Next, we implement the routine coefficients , which receives as input the triple (z1, z2, n) and returns a list with all the structure constants αi and αj−1 j involved in the law of the associated liform Lie algebra. > coefficients:=proc(z_1,z_2,n) 52 Chapter 3. Theoretical Study > local M; M:={}; for i from 1 to > z_2-z_1+1 do M:={op(M),a[i]}; od; > for j from 2 to 2*n-z_1-z_2-1 do > M:={op(M),a[j][j-1]}; od; > return M; end proc: > DGsetup(law(z_1,z_2,n)); From here on, we can operate over the Lie algebra associated with the output given by law . This algebra is denoted by Alg1 . Now, we change the format of the subindexes of the basis vectors with the command assign . > assign([seq(e||i=e[i],i=1..n)]); After that, we have to dene the variable Eq from the output of the routine law . Now, we execute this sentence Alg1 > L := LieAlgebraData(Eq,[seq(e[i],i=1..n)], Alg2); > DGsetup(L); From here on, we work over the Lie algebra Alg2 . With the command LieAlgebraData we can evaluate the following sentence Alg2 > TF, EQ, SOLN, AlgList :=Query(coefficients(z_1,z_2,n),"Jacobi"); This sentence provides us the conditions given by the Jacobi identities in terms of equations for the liform Lie algebra. Additionally, AlgList shows us a list with all the non-zero brackets. Finally, for each term given by the previous output, we have to write the following commands Alg2 > DGsetup(AlgList[i]); Alg2_i > f:=proc() if BracketOfSubspaces([ez_1],[en])=[] then Alg2_i > return "It is not a filiform Lie algebra"; end if; if Alg2_i > BracketOfSubspaces([ez_2],[ez_2+1])=[] then return Alg2_i > "It is not a filiform Lie algebra"; end if; else return Alg2_i > AlgList[i],SOLN[i]; fi; end proc: Alg2_i > f() With these sentences, we program a routine which studies if the Lie algebra given in AlgList[i] is compatible with the denition of the invariants z1 and z2 . In the armative case, the output is the set of conditions for the structure constants given by the routine coefficients and the non-zero brackets of the Lie algebra. In case of incompatibility, the output is the message It is not a liform Lie algebra". Filiform Lie algebras 53 Using this algorithm and its implementation, we have computed the triples (z1, z2, n) associated with non-model liform Lie algebras up to dimension 14 . This triples are shown in Tables 3.13.3, structured according to the maximal dimension for their abelian ideals and including the restrictions for the coecients αi and αj−1 j , if they exist. Next, we show a computational study of the algorithm. To do so, we have used an Intel Core 2 Duo T 5600 with a 1.83 GHz processor and 2.00 GB of RAM. Table 3.4 shows some computational data about both the computing time and the memory usage. These data were obtained considering the family of liform Lie algebras associated with the triple (n−3, n −3, n) . Additionally, we show some brief statistics about the relation between the computing time and the memory usage by the implementation. In this sense, Fig. 3.1 shows the behavior of both the computing time and the memory usage according to the dimension n . In fact, the computing time increases more quickly than the memory usage. As we can observe in Fig. 3.2, the increase of the computing time corresponds to a positive exponential model, whereas the memory usage does not follow such a model. Finally, we have also studied the quotients between memory usage and computing time, obtaining the frequency diagram shown in Fig. 3.3. In this case, the behavior can be also considered exponential, although this time is negative. 60 Chapter 4. Algorithm and applications mmands like Flatten and others related to Linear Algebra, since Lie algebras are vector spaces endowed with a second inner structure: the Lie bracket. Besides, the library combinat has to be also loaded to apply commands related to Combinatorial Algebra. Finally, we have also loaded the library Maplets[Elements] in order to display a message so that the user introduces the required input in the rst subroutine, which is devoted to dene the law of the Lie algebra considered. Now, we show the dierent steps constituting the algorithm and its corresponding implementation. The structure of the algorithm is based on two main routines calling several other subroutines with dierent functions. Let us note that all the routines are written in the same worksheet in order to run it after introducing the data asked for the dialog window built with the library Maplets[Elements] . 1. Implementing a subroutine which computes the Lie bracket between two arbitrary basis vectors in Bn . This subroutine depends on the law of g . The rst subroutine, named law , receives two natural numbers as inputs. These numbers represent the subindexes of two basis vectors in Bn . The subroutine returns the result of the bracket between these two vectors. Besides, conditional sentences are included to determine non-zero brackets (which are introduced in the subroutine) and the skew-symmetry property. Since the user has to complete the implementation of this subroutine with the non-zero brackets of g , we have also added a sentence at the beginning of the implementation, reminding this fact. Note that before running any other sentence, we must restart all the variables and delete all the computations saved for another law used before. Additionally, we must update the value of variable dim , which saves the dimension of the algebra to be studied. > restart: > maplet:=Maplet(AlertDialog("Don't forget to introduce non-zero brackets of the algebra and its dimension in subroutine law", 'onapprove'=Shutdown("Continue"),'oncancel'=Shutdown("Aborted"))): > Maplets[Display](maplet): > assign(dim,...): > law:=proc(i,j) > if i=j then return 0; end if; > if i>j then return -law(j,i); end if; > if (i,j)=... then return ...; end if; > if .... > else return 0; end if; > end proc; Algorithmic method 61 The ellipsis in command assign corresponds to write the dimension of the algebra g to be studied. The following two suspension points are associated with the computation of [Zi, Zj] : First, the value of the subindexes (i, j) and second, the result of [Zi, Zj] with respect to Bn . The last ellipsis denotes the rest of non-zero brackets. For each non-zero bracket, a new sentence if has to be included in the cluster. 2. Programming a subroutine to compute the bracket between two vectors expressed as a linear combination of vectors from the basis Bn of g . Then, we implement a subroutine, bracket , which computes the bracket between two arbitrary vectors of g . These vectors are expressed as linear combination of the vectors in Bn . The subroutine law is called in the implementation. > bracket:=proc(u,v,n) > local exp; exp:=0; > for i from 1 to n do > for j from 1 to n do > exp:=exp + coeff(u,Z[i])*coeff(v,Z[j])*law(i,j); > end do; > end do; > return exp; > end proc: 3. For each k -dimensional subalgebra h of g , computing the bracket between two arbitrary vectors in the basis of h . Those vectors are linear combinations of a main vector (whose coecient is equal to 1 ) and the n−k non-main ones. These expressions depend on the dimension of h . After introducing the law of g , we have to compute the brackets in an arbitrary subalgebra h . To do so, we implement the subroutine eq , which requires four inputs: the dimension n of g ; the subindexes i and l , indicating the main vectors in the bracket to be computed; and a list M with the subindexes of the non-main vectors in h . To do so, three local variables eqt , L and P are dened. For computing the brackets between the vectors in Bn , the subroutine eq calls the subroutine bracket , which is necessary to obtain each bracket in the law of h . Whereas the variable eqt saves the expression of the bracket belonging to the law of h , the list P takes the elements of M two by two and nally, L is a list containing all the coecients in the expression of eqt with respect to Bn . Precisely, the list L is the rst term of the output of the subroutine eq . The second is a list with the subindexes i and l corresponding to L . Let us note 62 Chapter 4. Algorithm and applications that the subindexes of the main vectors have to be saved together with the coecients in order to use them in a later subroutine. Each vector in the subalgebra h can be expressed as a linear combination of one main vector and the n−k non-main ones according to expression (1.4), where each row represents the coecients of one vector in the basis of h . Obviously, we can assume that the coecient of each main vector is equal to 1, because the row of (1.4) corresponding to that main vector can be divided by its coecient. To implement the subroutine eq , the coecients of the non-main vectors are denoted by b[i,k] . > eq:=proc(n,i,l,M::list) > local eqt,L,P; L:=[]; > if nops(M)=1 then P:=[[M[1],M[1]]] else P:=choose (M,2); end if; > eqt:=law(i,l); > for k from 1 to nops(M) do > eqt:=eqt + b[l,M[k]]*law(i,M[k]) + b[i,M[k]]*law(M[k],l); > end do; > for j from 1 to nops(P) do > eqt:=eqt+(b[i,P[j][1]]*b[l,P[j][2]]-b[i,P[j][2]]*b[l,P[j][1]])* law(P[j][1],P[j][2]); > end do; > for m from 1 to n do > L:=[op(L),coeff(eqt,Z[m])]; > end do; > return L,[i,l]; > end proc; Let us note that it is also possible to program the subroutine eq by using the subroutine bracket . However, we will consider the previous implementation for the computational study due to the fact that if we consider an implementation of eq which calls the subroutine bracket , both the computing time and the used memory will increase. > eq:=proc(n,i,l,M::list) > local eqt,L,u,v; > L:=[]; eqt:=0; u:=Z[i]; v:=Z[l]; > for k from 1 to nops(M) do > u:=u+b[i,M[k]]*Z[M[k]]; v:=v+b[l,M[k]]*Z[M[k]]; > end do; > eqt:=bracket(u,v,n); > for m from 1 to n do > L:=[op(L),coeff(eqt,Z[m])]; Algorithmic method 63 > end do; > return L,[i,l]; > end proc: 4. Solving a system whose equations are obtained by imposing the abelian law to the brackects computed in the previous step for the subalgebra h . Next, we implement the subroutine sys , which receives two inputs: The dimension n of g and a list M with the subindexes of the non-main vectors in the basis of h . This subroutine solves the system of equations generated by the subroutine eq . Four local variables L , P , R and S have been dened for its implementation: L is a list with the subindexes of the main vectors; the list R contains the expressions computed by the subroutine eq ; P is dened as in the previous subroutine; and, nally, S is a set where the equations of the system are saved > sys:=proc(n,M::list) > local L,P,R,S; L:=[]; R:=[]; S:={}; > for x from 1 to n do > if member(x,convert(M,set))=false then L:=[op(L),x]; end if; > end do; > if nops(L)=1 then P:=[[L[1],L[1]]] else P:=choose (L,2); end if; > for j from 1 to nops(P) do > r[j]:=[eq(n,P[j][1],P[j][2],M)]; > end do; > R:=[seq(r[i][1],i=1..nops(P))]; > for y from 1 to nops(R) do > for k from 1 to n do > S:={op(S),R[y][k]=0}; > end do; > end do; > return {solve(S)}; > end proc; 5. Programming a subroutine which determines the existence of abelian subalgebras in a xed dimension. This subroutine, called absub , is implemented by introducing two natural numbers n and k , namely: n is the dimension of g and k is less than n . This subroutine determines the existence of abelian subalgebras with dimension k . Two local variables are used by the subroutine: L and S . The rst variable, L , is a list whose elements are lists with the subindexes of the n − k non-main 64 Chapter 4. Algorithm and applications vectors. The variable S is a set with the solutions given by the subroutine sys . In this way, absub returns a message indicating the non-existence of k -dimensional abelian subalgebras or, if there exist k -dimensional abelian subalgebras, returns the set S . Since the coecient of each main vector is 1, the system given by the subroutine sys has not solutions when S vanishes. When the system has some solution, the family of computed vectors is linearly independent and forms a basis of the subalgebra. Let us note that, the solutions in S will be determined according to the the eld K that we are considering. For example, if the solutions in S contain complex coecients, there will be no real solutions for the system solved by sys and there do not exist any abelian subalgebras of dimension k for the case K=R . For this eld, it would be necessary to include a conditional sentence for determining if such complex coecients appear. > absub:=proc(n,k) > local L,S; L:=choose(n,n-k); S:={ }; > for i from 1 to nops(L) do > if sys(n,L[i])={{}} then S:=S else > for j from 1 to nops(sys(n,L[i])) do > S:={op(S),{convert(L[i],set),sys(n,L[i])[j]}}; > end do; > end if; > end do; > if S={} then return "There is no abelian subalgebra"; end if; > if S={{}} then return "There is no abelian subalgebra" else return S; > end if; > end proc; 6. Computing α(g) by ruling out dimensions for abelian subalgebras. Next, we implement the subroutine alpha , which receives the dimension n of g as its unique input and returns the α invariant of g . The subroutine starts studying if α(g) = n by using the subroutine absub . Then, a loop is programmed to stop when absub does not nd abelian subalgebras. > alpha:=proc(n) > if type(absub(n,n-1),set)=true then return n-1; end if; > for i from 2 to n-1 do > if absub(n,i)="There is no abelian subalgebra" then return i-1; > end if; > end do; > end proc; Algorithmic method 65 7. Computing the basis of an abelian subalgebra of maximal dimension, that is, a subalgebra with dimension α(g) . The following subroutine, named asmd , receives as input the dimension n of g and returns the basis of an abelian subalgebra of maximal dimension. To do so, this subroutine calls the subroutines alpha and absub . > asmd:=proc(n) > local u,L,R,S,B,k; > k:=alpha(n);S:={};L:={}; u:=absub(n,k); > if k=1 then return {seq({Z[i]},i=1..n)}; end if; > if type(u[1][1],set(integer))=true > then R:=u[1][1]; S:=u[1][2] else > R:=u[1][2]; S:=u[1][1]; > end if; > for x from 1 to n do > if member(x,R)=false then L:={op(L),x}; > end if; > end do; > for i from 1 to nops(L) do > b[i]:=Z[L[i]]; end do; > for i from 1 to nops(L) do > for j from 1 to nops(R) do > b[i]:=b[i]+a[L[i],R[j]]*Z[R[j]]; > end do; > end do; > B:={seq(b[i],i=1..nops(L))}; > return eval(B,S); > end proc: 8. Computing the basis of an abelian subalgebra for a xed set of non-main vectors and some restrictions given by the previous subroutines. Now, we implement the subroutine basabsub , which receives three inputs: the dimension n of g and two sets, S and T , with the subindexes of the non-main vectors in the basis of h . We will use this subroutine with the solution given by sys . We have dened four local variables R , B , M and N for its implementation. First, we introduce a conditional sentence if for the sets M and N in the cluster to nd out whether S or T is the set of non-main vectors. This is due to the fact that MAPLE 12 sometimes returns the solutions in dierent order. R is a set with the subindexes of the main vectors and, in the set B , we compute the basis for the abelian subalgebra. In this way, B is the output of this subroutine. 66 Chapter 4. Algorithm and applications > basabsub:=proc(n,S::set,T::set) > local R,B,M,N; R:={};B:={}; > if type(S,set(integer))=true then M:=S; N:=T else M:=T; N:=S; end if; > for x from 1 to n do > if member(x,M)=false then R:={op(R),x}; end if; > end do; > for i from 1 to nops(R) do > a[i]:=Z[R[i]]; > end do; > for i from 1 to nops(R) do > for j from 1 to nops(M) do > a[i]:=a[i] + b[R[i],M[j]]*Z[M[j]]; > end do; > end do; > B:={seq(a[i],i=1..nops(R))}; > return eval(B,N); > end proc: 9. Programming a subroutine which computes a list with all the abelian subalgebras of g with certain dimension k . The following subroutine, named listabsub , requires two inputs: the dimension n of g and a natural number k , less than n and which corresponds with the dimension of the abelian subalgebra. To implement it, we consider two local variables S and L . This subroutine calls the subroutine basabsub to compute a basis for each k -dimensional abelian subalgebra. Whereas this value is saved in the local variable S , L is a set with the bases of each abelian subalgebra of g with dimension k . Precisely, the list L is the output of the subroutine listabsub . > listabsub:=proc(n,k) > local S,L; S:=absub(n,k);L:={}; > if k=1 then return {seq({Z[i]},i=1..n)}; end if; > if S="There is no abelian subalgebra" then return {}; end if; > for i from 1 to nops(S) do > L:={op(L),basabsub(n,S[i][1],S[i][2])}; > end do; > return L; > end proc: Let us note that it is also possible to give an equivalent implementation for the subroutine asmd by using the subroutine listabsub . Algorithmic method 67 > asmd:=proc(n) > local k; > k:=alpha(n); > return listabsub(n,k); > end proc: 10. Implementing the routine to compute a list with the bases of all the non-trivial abelian subalgebras of g by using the previous subroutines. Now, we implement the routine allabsub , which receives the dimension n of g as its unique input. The routine allabsub returns a set with the bases of all the abelian subalgebras of g with dimension less than or equal to α(g) . In this way, the routine starts computing α(g) and then, the output is dened by using the previous subroutine listabsub . > allabsub:=proc(n) > local B,k; k:=alpha(n);B:={}; > for i from 1 to k-1 do > B:={op(B),listabsub(n,i)}; > end do; > return B; > end proc: 11. Programming a subroutine which determines if there is an abelian ideal associated with a given abelian subalgebra. Next, we explain the subroutine abideal , which requires two inputs: a set S with the basis of an abelian subalgebra and the dimension n of g . The subroutine determines the existence of an abelian ideal from the basis S of an abelian subalgebra, obtained with the subroutine listabsub for a xed dimension. To do so, we impose that S has to be the basis of an abelian ideal. Then, we solve the system: if there is no solution, the output of this subroutine is the message It is not an abelian ideal" and if there is a solution, it returns the basis of an abelian ideal. > abideal:=proc(S,n) > local w, R, L, Q, M; w:=0; R:=[]; L:=[]; Q:={}; M:={}; N:={}; > for i from 1 to nops(S) do > w:=w + a[i]*S[i]; > end do; > for i from 1 to nops(S) do > for j from 1 to n do 68 Chapter 4. Algorithm and applications > if bracket(S[i],Z[j],n)<>0 then > L:=[op(L),bracket(Z[j],S[i],n)]; else L:=L; end if; > end do; > end do; > for i from 1 to nops(L) do r[i]:=0; > for j from 1 to nops(S) do > r[i]:=r[i]+c[i,j]*S[j]; > end do; > end do; > R:=[seq(r[i],i=1..nops(L))]; > M:={seq(L[k]-R[k], k=1..nops(L))}; > for i from 1 to nops(M) do > Q:={op(Q),seq(coeff(M[i],Z[j])=0,j=1..n)}; > end do; > if {solve(Q)}={} then return "It is not an abelian ideal" else > return eval(S,solve(Q)); > end if; > end proc: 12. Computing β(g) from α(g) and the previous subroutine. The subroutine beta receives the dimension n of g as its unique input and returns the β invariant of g . Let us note that this value can be zero (semisimple Lie algebras). The subroutine starts computing the value of α . Then, a loop is programmed by using the previous subroutine and listabsub > beta:=proc(n) > local r; r:=alpha(n); > for k from 0 to r-1 do > for i from 1 to nops(listabsub(n,r-k)) do > if abideal(listabsub(n,r-k)[i],n)<>"There is no abelian ideal" > then return r-k; > end if; > end do; > end do; > return 0; > end proc: 13. Implementing a subroutine which determines the set of abelian ideals of maximal dimension, that is, abelian ideals with dimension β(g) . Next, in this subroutine, named aimd , we compute the basis of an abelian ideal of maximal dimension; that is, an abelian ideal with dimension β(g) . To do so, the routine aimd calls the subroutines beta , listabsub and abideal . First, Algorithmic method 69 we compute the set of all abelian subalgebras of dimension β(g) and then we apply the subroutine abideal to obtain abelian ideals. > aimd:=proc(n) > local k,S,T; k:=beta(n);S:=listabsub(n,k);T:={}; > for i from 1 to nops(S) do > T:={op(T),abideal(S[i],n)}; > end do; > return T; > end proc: 14. Programming the routine to compute a list with the bases of all the non-trivial abelian ideals of g by using the previous subroutines. The routine allabideal receives the dimension n of g as its unique input. This routine returns a set with the basis of all the abelian ideals of g with dimension less than or equal to β(g) . The output of this routine is dened by using the subroutines listabsub and abideal . > allabideal:=proc(n) > local B, k; k:=beta(n); B:={}; > if k=0 then return {}; else > for i from 1 to k do > for j from 1 to nops(listabsub(n,i)) do > if abideal(listabsub(n,i)[j],n)<>"There is no abelian ideal" > then B:={op(B),abideal(listabsub(n,i)[j],n)}; > end if; > end do; > end do; > end if; > return B; > end proc: Now, we show an example using the 4 -dimensional Lie algebra with non-zero brackets [Z1, Z2] = Z3 , [Z1, Z3] = Z4 . First, we have to complete the implementation of the subroutine law as follows > restart: > maplet:=Maplet(AlertDialog("Don't forget to introduce non-zero brackets of the algebra and its dimension in subroutine law", 'onapprove'=Shutdown("Continue"), 'oncancel'=Shutdown("Aborted"))): > Maplets[Display](maplet): > assign(dim,4): 76 Chapter 4. Algorithm and applications Proof. The 4 -dimensional subalgebra of g5,j generated by ⟨e1, e2, e3, e4⟩ is abelian for j∈ {7,...,18} . Since g5,j is not abelian, α(g5,j) = 4 and the previous subalgebra is an abelian subalgebra of maximal dimension. Moreover, D(g5,j) is a 4 -dimensional abelian ideal for j∈ {7, . . . , 18} . Consequently, these algebras are 2 -step solvable. The subalgebras ⟨e1, e3, e4⟩ and ⟨e1, e2, e3⟩ are abelian subalgebras of g5,j for j∈ {19,...,29} and for j∈ {30, . . . , 35,38,39} , respectively. So, we can set that α(g5,j)≥3 , for j∈ {19, . . . 35,38,39} . Consequently, it is sucient to prove the non-existence of 4 -dimensional abelian subalgebras of g5,j . Once more, the reasoning is analogous for all these algebras. So we only study explicitly the algebra g5,22 , whose law, with respect to a certain basis {ei}5 i=1 , is [e2, e3] = e1,[e2, e5] = e3,[e4, e5] = e4 . By applying the reasoning and notation used in Proposition 4.1, assume the existence of a 4 -dimensional abelian subalgebra and nd a non-zero bracket in its law. • For k∈ {1,4,5} : [e2+λ2ek, e3+λ3ek] = e1+v is non-zero, because v∈ ⟨{ei}5 i=2⟩ . • For k∈ {2,3} : [e4+λ4ek, e5+λ5ek] = e4+w is non-zero, because w∈ ⟨e1, e2, e3, e5⟩ . So, there do not exist 4 -dimensional abelian subalgebras of g5,22 and α(g5,22) = 3 . The subalgebra ⟨e1, e2⟩ is abelian for g5,36 and g5,37 . Hence, both α(g5,36) and α(g5,37)≥2 and we only have to prove the non-existence of abelian subalgebras of dimension 3 . Both cases are analogous, so we will prove it for g5,37 . Let us suppose the existence of a 3 -dimensional abelian subalgebra. We can express 3 -dimensional subalgebras as follows aj,k =⟨{ei+λiej+µiek|1≤i≤5∧j, k =i}⟩, where ej and ek are the two non-main vectors. Proving that aj,k is non-abelian is equivalent to nding a non-zero bracket in its law. • For (j, k)∈ {(1,2),(1,4)} : The bracket [e3+λ3ej+µ3ek, e5+λ5ej+µ5ek] = e2+v is non-zero, because v∈ ⟨e1, e3, e4, e5⟩ . • For (j, k)∈ {(1,3),(1,5)} : The bracket [e2+λ2ej+µ2ek, e4+λ4ej+µ4ek] = e2+v is non-zero, because v∈ ⟨e1, e3, e4, e5⟩ . • For (j, k)∈ {(2,5),(3,5)} : The bracket [e1+λ1ej+µ1ek, e4+λ4ej+µ4ek] = 2e1+w is non-zero, because w∈ ⟨ei⟩5 i=2 . α and β invariants for low-dimensional Lie algebras 77 • For (j, k) = (4,5) : The bracket [e2+λ2ej+µ2ek, e3+λ3ej+µ3ek] = e1+w is non-zero, because w∈ ⟨ei⟩5 i=2 . • For (j, k) = (2,3) : If a2,3 is abelian, the bracket [e1+λ1e2+µ1e3, e4+λ4e2+µ4e3] = (2 + λ1µ4−µ1λ4)e1+λ1e2+µ1e3 would be zero, obtaining the system {2 + λ1µ4−µ1λ4= 0, λ1=µ1= 0} , which has no solution. • For (j, k) = (2,4) : Consider the brackets [e3+λ3e2+µ3e4, e5+λ5e2+µ5e4] = −λ5e1+(1+λ3µ5−µ3λ5)e2+(µ5−λ3)e3, [e1+λ1e2+µ1e4, e5+λ5e2+µ5e4] = 2µ5e1−λ1e3+ (λ1µ5−µ1λ5)e2. If a2,4 is abelian, these brackets would be zero and the system {1+λ3µ5−µ3λ5= 0, µ5=λ1=λ5=0, µ5−λ3=0, λ1µ5−µ1λ5= 0} without solutions would be obtained. • For (j, k) = (3,4) : Consider the brackets [e2+λ2e3+µ2e4, e5+λ5e3+µ5e4] = λ5e1+ (λ2µ5−µ2λ5−1)e3+ (µ5+λ2)e2, [e1+λ1e3+µ1e4, e2+λ2e3+µ2e4] = −µ1e2+ (2µ2−λ1)e1+ (λ1µ2−µ1λ2)e3, [e1+λ1e3+µ1e4, e5+λ5e3+µ5e4] = 2µ5e1+λ1e2+ (λ1µ5−µ1λ5)e3. If a3,4 is abelian, these brackets would be zero, obtaining the following system without solutions {λ2µ5−µ2λ5−1= 0, µ1=µ5=λ1=λ5= 0, µ5+λ2= 0,2µ2−λ1= 0, λ1µ2−µ1λ2= 0, λ1µ5−µ1λ5= 0} . Hence, there are no 3 -dimensional abelian subalgebras in g5,37 and α(g5,37) = 2 . Remark 4.1. Regarding Lie algebras g5,25,g5,26 and g5,37 , we have an analogous situation to the one shown in Example 2.2. This is due to the fact that over R , α(g5,25) = α(g5,26)=3 , but β(g5,25) = β(g5,26) = 2 since the abelian ideals of dimension 3 for both algebras are ⟨e1, e4, e3±ie2⟩ and ⟨e1, e4, e2±ie3⟩ . For Lie algebra g5,37 we have α(g5,37) = 2 , but β(g5,37) = 1 since the 2 -dimensional abelian ideals of g5,37 are ⟨e1, e2±ie3⟩ and ⟨e1, e3±ie2⟩ . Proposition 4.3. Let g be a 6 -dimensional non-decomposable real solvable nonnilpotent Lie algebra. Then, the possible values of α(g) are given in Tables 4.84.14. 78 Chapter 4. Algorithm and applications Proof. In order to prove this result, we can use an analogous reasoning to the one used in the proof of Proposition 4.2 bearing in mind that • ⟨e3, e4, e5, e6⟩ is an abelian subalgebra of g6,j , for j= 1, . . . , 27 . • ⟨e3, e4, e5⟩ and ⟨e3, e4, e6⟩ are abelian subalgebras of g6,28 and g6,j , for j= 29, . . . 40 , respectively. • ⟨e1, e2, e3, e4, e5⟩ is an abelian subalgebra of g6,j , for j= 41, . . . , 52 . • ⟨e1, e2, e4, e5⟩ is an abelian subalgebra of g6,j , for j= 53, . . . , 78 . • ⟨e1, e2, e3, e4⟩ is an abelian subalgebra of g6,j , for j= 79, . . . , 115 . • ⟨e1, e2, e3⟩ is an abelian subalgebra of g6,j , for j= 116,...,139 . Remark 4.2. Table 4.7 shows the cases of 6 -dimensional non-decomposable real solvable Lie algebras where α and β invariants are dierent, due to the fact that the abelian ideals of dimension α are dened over C . 4.2.3 Nilpotent Lie algebras of dimension less than 8 The invariant α(g) for a complex nilpotent Lie algebra g has been obtained up to dimension 6 in connection with degenerations [19, 92]. We want to give a list here, thereby correcting a few typos in [92]. In dimension 7 there is no list for α(g) , as far as we know. We use the classication of complex nilpotent Lie algebras up to dimension 7 by Magnin [71], and for dimension 6 also by de Graaf [33] and Seeley [92]. The result for the indecomposable algebras in dimension n≤5 is as follows Proposition 4.4. Let g be a complex nilpotent Lie algebra of dimension less than 6 . Then, the possible values of α(g) are given in Table 4.15. Proof. The α invariant of Lie algebras n3 and n4 was computed and pointed out in Table 4.4 . First, we prove that α(g5,2) = α(g5,5) = 4 . Let us note that in this case, ⟨e2, e3, e4, e5⟩ is an abelian subalgebra of g5,2 and g5,5 , that are non-abelian, so α(g5,2) = α(g5,5) = 4 . Now, we compute a 3 -dimensional abelian subalgebra of g5,i , ∀i∈ {1,3,4,6} , and we have to prove that it is not possible to obtain an abelian subalgebra of them with dimension greater than 3 . The algebra ⟨e3, e4, e5⟩ is an abelian subalgebra of g5,i , for i∈ {1,3,4,6} . Consequently, it is sucient to prove that it is not possible to obtain a 4 -dimensional abelian subalgebra of g5,i , for α and β invariants for low-dimensional Lie algebras 79 i∈ {1,3,4,6} . For reasons of length, we only study explicitly one example of these algebras. The other cases are analogous. Let us consider Lie algebra g5,3 which law with respect to the basis {ei}5 i=1 is given by: [e2, e4] = e3,[e2, e5] = e1,[e4, e5] = e2 . Let us suppose that there exists a 4 -dimensional abelian subalgebra in g5,3 . This subalgebra can be expressed as ak=⟨{ei+λiek|1≤i≤5∧i=k}⟩, where ek is the non-main vector. To prove that ak is non-abelian it is necessary and sucient to nd a non-zero bracket in its law. • For k∈ {1,3,5} , the following bracket is non-zero: [e2+λ2ek, e4+λ4ek] = e3+v, because the vector v belongs to ⟨e1, e2, e4, e5⟩ . • For k= 2 , a non-zero bracket is given by [e4+λ4e2, e5+λ5e2] = e2+v, because the vector v belongs to ⟨e1, e3, e4, e5⟩ . • For k= 4 , the following bracket is non-zero: [e2+λ2e4, e5+λ5e4] = e1+v, since v belongs to ⟨{ei}i=5 i=2⟩ . Consequently, there do not exist 4 -dimensional abelian subalgebras in g5,3 and α(g5,3) = 3 . Proposition 4.5. Let g be a nilpotent Lie algebra of dimension 6 . Then, the possible values of α(g) are given in Table 4.16. Proof. Clearly α(C6)=6 . Since α invariant is additive in virtue of Lemma 2.1, its value for Lie algebras g5,i ⊕C , for i= 1,2, . . . , 6 ; n3⊕n3 ; n3⊕C3 and n4⊕C2 can be computed directly. Consequently, we only have to compute the value of α for Lie algebras g6,i , for i∈ {1,...,20} . These algebras are shown in Table 4.17. Let us note that ⟨e1, e3, e4, e5, e6⟩ and ⟨e2, e3, e4, e5, e6⟩ are 5 -dimensional abelian subalgebras of g6,6 and g6,16 , respectively. Since these algebras are non-abelian, we can conclude that α(g6,6) = α(g6,16) = 5 . Moreover, ⟨e3, e4, e5, e6⟩ , ⟨e2, e4, e5, e6⟩ and ⟨e1, e4, e5, e6⟩ are abelian subalgebras of g6,i , for i∈ {1,3,4,5,7,8,11,12,14,15,17,19} ; g6,j , for j∈ {2,10,13} and g6,9 , respectively. So, α invariant of these algebras is, at least, 4 . In this way, we have to see if there exists some abelian subalgebra of them with dimension 5 . We study one example of the Lie algebras considered and the other are analogous. We consider the algebra g6,19 =⟨e1, e2, e3, e4, e5, e6⟩ with law [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6,[e2, e3] = e5,[e2, e4] = e6 . A 5 -dimensional abelian subalgebra is expressed by ak as in Proposition 4.4. 80 Chapter 4. Algorithm and applications • For k∈ {3, . . . , 6} , the bracket [e1+λ1ek, e2+λ2ek] = e3+v, is non-zero since v belongs to ⟨{ei}i=3⟩ . • For k= 2 , a non-zero bracket is [e1+λ1Z2, e3+λ3Z2] = e4+λ3e3+λ1e5 . • For k= 1 , this bracket is non-zero: [e2+λ2e1, e3+λ3e1] = e5−λ3e3+λ2e4. There is no 5 -dimensional abelian subalgebra in g6,19 , so α(g6,19) = 4 . The Hasse diagram for degenerations of nilpotent Lie algebras in dimension 6 is shown in Figure 4.4, removing the typos in [92]. If g→deg h , then α(g)≤α(h) . In dimension 7 , we use Magnin's classication of non-decomposable, complex nilpotent Lie algebras in [71] (see Table 4.184.21) to compute the α invariant of each isomorphism class. Note that 4≤α(g)≤6 in this case according to Lemma 2.5 . The result is as follows Proposition 4.6. Let g be a non-decomposable nilpotent Lie algebra of dimension 7 . Then, the possible values of α(g) are the following α(g) = 4 : g=g7,0.1,g7,0.4(a),g7,0.5,g7,0.6,g7,0.7,g7,0.8,g7,1.02,g7,1.03,g7,1.1(ia),a=1, g7,1.1(ii),g7,1.1(iii),g7,1.1(iv),g7,1.1(v),g7,1.1(vi),g7,1.2(ia),a=1,g7,1.2(ii), g7,1.2(iii),g7,1.2(iv),g7,1.3(ia),a=0,g7,1.3(ii),g7,1.3(iii),g7,1.3(iv),g7,1.3(v), g7,1.5,g7,1.8,g7,1.11,g7,1.14,g7,1.17,g7,1.19,g7,1.20,g7,1.21,g7,2.1(ia),a=0,1, g7,2.1(ii),g7,2.1(iii),g7,2.1(iv),g7,2.1(v),g7,2.2,g7,2.4,g7,2.5,g7,2.6,g7,2.10, g7,2.12,g7,2.13,g7,2.17,g7,2.23,g7,2.26,g7,2.28,g7,2.29,g7,2.30,g7,2.34, g7,2.35,g7,2.37,g7,3.1(ia),a=0,1,g7,3.1(iii),g7,3.13,g7,3.18,g7,3.22,g7,4.4. α(g) = 5 : g=g7,0.2,g7,0.3,g7,1.01(i),g7,1.01(ii),g7,1.1(ia),a=1,g7,1.2(ia),a=1,g7,1.3(ia),a=0, g7,1.4,g7,1.6,g7,1.7,g7,1.9,g7,1.10,g7,1.12,g7,1.13,g7,1.15,g7,1.16,g7,1.18, g7,2.1(ia),a=0,1,g7,2.7,g7,2.8,g7,2.9,g7,2.11,g7,2.14,g7,2.15,g7,2.16,g7,2.18, g7,2.19,g7,2.20,g7,2.21,g7,2.22,g7,2.24,g7,2.25,g7,2.27,g7,2.31,g7,2.32, g7,2.33,g7,2.36,g7,2.38,g7,2.39,g7,2.40,g7,2.41,g7,2.42,g7,2.43,g7,2.44, g7,2.45,g7,3.1(ia),a=0,1,g7,3.3,g7,3.4,g7,3.5,g7,3.6,g7,3.7,g7,3.8,g7,3.9, g7,3.10,g7,3.11,g7,3.12,g7,3.14,g7,3.15,g7,3.16,g7,3.17,g7,3.19,g7,3.21, g7,3.23 g7,3.24,g7,4.1,g7,4.3. α(g) = 6 : g=g7,2.3,g7,3.2,g7,3.20,g7,4.2. α and β invariants for low-dimensional Lie algebras 81 Figure 4.4: Hasse diagram for degenerations of nilpotent Lie algebras in dimension 6. g6,20 tthhhhhhhhhhhhhhhhhhhhhh ** V V V V V V V V V V V V V V V V V V V V V V g6,18  )) S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S g6,19  qqdddddddddddddddddddddddddddddddddddddddddddddddd tthhhhhhhhhhhhhhhhhhhhhh g6,17 && L L L L L L L L L L )) R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R ** V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V Vg6,15 xxrrrrrrrrrr && M M M M M M M M M M ** V V V V V V V V V V V V V V V V V V V V V Vg6,13 xxqqqqqqqqqq g6,16  g6,14 yyrrrrrrrrrr  g6,9  g6,12 qqdddddddddddddddddddddddddddddddddddddddddddddd rreeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee xxrrrrrrrrrrr  g5,6⊕C ** U U U U U U U U U U U U U U U U U U U U Ug6,5 %% K K K K K K K K K K K ,, X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Xg6,10  %% L L L L L L L L L L Lg6,11 qqdddddddddddddddddddddddddddddddddddddddddddddd }}| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |   g5,5⊕C  g6,8 yyrrrrrrrrrrr  g6,4 rreeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee }}{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{ g6,7 yyrrrrrrrrrrr  , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,, Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Yg6,2  }}|||||||||||||||||||||||||||||| && L L L L L L L L L L g6,6  : : : : : : : : : : : : : : : :g5,4⊕C   g5,3⊕C uullllllllllllllllllllllllllllllllll  n3⊕n3 %% L L L L L L L L L L L n4⊕C2  9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9g6,1   g6,3  g5,2⊕C  g5,1⊕C yyrrrrrrrrr n3⊕C3  C6 82 Chapter 4. Algorithm and applications 4.2.4 Tables Table 4.4: Lie algebras of dimension less than 5 gdim(g) Lie brackets α(g) g1,1=C11 1 g2,1=C22 2 g2,2=r22 [e1, e2] = e11 g3,1=C33 3 g3,2=n33 [e1, e2] = e32 g3,3=r2⊕C13 [e1, e2] = e12 g3,4=r33 [e1, e2] = e2,[e1, e3] = e2+e32 g3,5=r3,λ 3 [e1, e2] = e2,[e1, e3] = λe3, λ ∈C∗,|λ| ≤ 1 2 g3,6=sl23 [e1, e2] = e3,[e1, e3] = −2e1,[e2, e3]=2e21 g4,1=C44 4 g4,2=n3⊕C14 [e1, e2] = e33 g4,3=r2⊕C24 [e1, e2] = e13 g4,4=r3⊕C14 [e1, e2] = e2,[e1, e3] = e2+e33 g4,5=r3,λ ⊕C14 [e1, e2] = e2,[e1, e3] = λe3, λ ∈C,0<|λ| ≤ 1 3 g4,6=r2⊕r24 [e1, e2] = e1,[e3, e4] = e32 g4,7=sl2⊕C14 [e1, e2] = e3,[e1, e3] = −2e1,[e2, e3]=2e22 g4,8=n44 [e1, e2] = e3,[e1, e3] = e43 g4,9=r4,α 4 [e1, e2] = e2,[e1, e3] = e3,[e1, e4] = αe4, α ∈C∗3 g4,10 =r4,α,β 4[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = αe2 −βe3+e4, α ∈C∗ , β∈C or α, β = 0 3 gα 4,11 4[e1, e2] = e3,[e1, e3] = e4, [e1, e4] = α(e2+e3), α ∈C∗3 g4,12 4 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e23 g4,13 4 [e1, e2] = 1 3e2+e3,[e1, e3] = 1 3e3,[e1, e4] = 1 3e43 g4,14 4[e1, e2] = e2,[e1, e3] = e3, [e1, e4] = 2e4,[e2, e3] = e4 2 g4,15 4 [e1, e2] = e3,[e1, e3] = e2,[e2, e3] = e42 gα 4,16 4[e1, e2] = e3,[e1, e3] = −αe2+e3, [e1, e4] = e4,[e2, e3] = e4, α ∈C2 α and β invariants for low-dimensional Lie algebras 83 Table 4.5: 5 -dimensional non-decomposable real solvable non-nilpotent Lie algebras g Lie brackets α(g) Parameters g5,7[e1, e5] = e1,[e2, e5] = αe2, [e3, e5] = βe3,[e4, e5] = γe4 4−1≤γ≤β≤α≤1 , αβγ = 0 . g5,8[e2, e5] = e1,[e3, e5] = e3,[e4, e5] = γe4,4 0 <|γ| ≤ 1 g5,9[e1, e5] = e1,[e2, e5] = e1+e3, [e3, e5] = βe3,[e4, e5] = γe4 4 0 =γ≤β g5,10 [e2, e5] = e1,[e3, e5] = e2,[e4, e5] = e44 g5,11 [e1, e5] = e1,[e2, e5] = e1+e2, [e3, e5] = e2+e3,[e4, e5] = γe4 4γ= 0 g5,12 [e1, e5] = e1,[e2, e5] = e1+e2, [e3, e5] = e2+e3,[e4, e5] = e3+e4 4 g5,13 [e1, e5] = e1,[e2, e5] = γe2 , [e3, e5] = pe3−se4,[e4, e5] = se3+pe4 4γs = 0,|γ| ≤ 1 g5,14 [e2, e5] = e1,[e3, e5] = pe3−e4 , [e4, e5] = e3+pe4 4 g5,15 [e1, e5] = e1,[e3, e5] = γe3, [e2, e5] = e1+e2,[e4, e5] = e3+γe4 4−1≤γ≤1 g5,16 [e1, e5] = e1,[e2, e5] = e1+e2, [e3, e5] = pe3−se4,[e4, e5] = se3+pe4 4s= 0 g5,17 [e1, e5] = pe1−e2,[e2, e5] = e1+pe2, [e3, e5] = qe3−se4,[e4, e5] = se3+qe4 4s= 0 g5,18 [e3, e5] = e1+pe3−e4,[e2, e5] = e1+pe2 [e1, e5] = pe1−e2,[e4, e5] = e2+e3−pe4 4p≥0 g5,19 [e2, e3] = e1,[e1, e5] = (1 + α)e1, [e2, e5] = e2,[e3, e5] = αe3,[e4, e5] = βe4 3β= 0 g5,20 [e2, e3] = e1,[e1, e5] = (1 + α)e2, [e2, e5] = e2,[e3, e5] = αe3, [e4, e5] = e1+ (1 + α)e4 3 g5,21 [e2, e3] = e1,[e1, e5] = 2e1,[e4, e5] = e4, [e2, e5] = e2+e3,[e3, e5] = e3+e4 3 g5,22 [e2, e3] = e1,[e2, e5] = e3, [e4, e5] = e4 3 g5,23 [e2, e3] = e1,[e1, e5] = 2e1,[e3, e5] = e3, [e2, e5] = e2+e3,[e4, e5] = βe4 3β= 0 g5,24 [e2, e3] = e1,[e1, e5] = 2e1,[e3, e5] = e3, [e2, e5] = e2+e3,[e4, e5] = ϵe1+ 2e4 3ϵ=±1 g5,25 [e2, e3] = e1,[e1, e5]=2pe1,[e4, e5] = βe4, [e2, e5] = pe2+e3,[e3, e5] = −e2+pe3 3β= 0 g5,26 [e2, e5] = pe2+e3,[e1, e5]=2pe1 , [e2, e3] = e1,[e3, e5] = −e2+pe3, [e4, e5] = ϵe1+ 2pe4 3ϵ=±1 g5,27 [e2, e3] = e1,[e3, e5] = e3+e4, [e1, e5] = e1,[e4, e5] = e1+e4 3 g5,28 [e2, e3] = e1,[e2, e5] = αe2,[e4, e5] = e4, [e1, e5] = (1 + α)e1,[e3, e5] = e3+e4 3 g5,29 [e2, e3] = e1,[e1, e5] = e1, [e2, e5] = e2,[e3, e5] = e4 3 g5,30 [e2, e4] = e1,[e3, e4] = e2, [e1, e5] = (2 + h)e1,[e4, e5] = e4 , [e2, e5] = (1 + h)e2,[e3, e5] = he3 3 84 Chapter 4. Algorithm and applications Table 4.6: 5 -dimensional non-decomposable real solvable non-nilpotent Lie algebras (II) g Lie brackets α(g) Parameters g5,31 [e2, e4] = e1,[e3, e4] = e2, [e1, e5]=3e1,[e3, e5] = e3, [e2, e5]=2e2,[e4, e5] = e3+e4 3 g5,32 [e2, e4] = e1,[e3, e4] = e2,[e1, e5] = e1, [e2, e5] = e2,[e3, e5] = he1+e3 3 g5,33 [e1, e4] = e1,[e3, e4] = βe3, [e2, e5] = e2,[e3, e5] = γe3 3β2+γ2= 0 g5,34 [e1, e4] = αe1,[e2, e4] = e2, [e3, e4] = e3,[e1, e5] = e1,[e3, e5] = e2 3 g5,35 [e1, e4] = he1,[e2, e4] = e2,[e3, e4] = e3, [e2, e5] = −e3,[e1, e5] = αe1,[e3, e5] = e2 3h2+α2= 0 g5,36 [e2, e3] = e1,[e1, e4] = e1, [e2, e4] = e2,[e3, e5] = e3,[e2, e5] = −e2 2 g5,37 [e2, e3] = e1,[e1, e4]=2e1,[e2, e4] = e2, [e3, e4] = e3,[e2, e5] = −e3,[e3, e5] = e2 2 g5,38 [e1, e4] = e1,[e2, e5] = e2,[e4, e5] = e33 g5,39 [e1, e4] = e1,[e2, e4] = e2, [e1, e5] = −e2,[e2, e5] = e1,[e4, e5] = e3 3 Table 4.7: Cases when α and β invariants are dierent in non-decomposable real solvable Lie algebras of dimension 6 gα(g)β(g) Abelian ideals of dimension α(g) over C g6,j , for j={73,74,77,78}4 3 ⟨e1, e4, e5, e2±ie3⟩ , ⟨e1, e4, e5, e3±ie2⟩ g6,k , for k={35,36,37,39,40}3 2 ⟨e3, e6, e4±ie5⟩ , ⟨e3, e6, e5±ie4⟩ g6,129 3 2 ⟨e1, e2, e3±ie5⟩ , ⟨e1, e2, e5±ie3⟩ ⟨e1, e4, e3±ie5⟩ , ⟨e1, e4, e5±ie3⟩ g6,130 3 2 ⟨e1, e2±e4, e3±ie5⟩ , ⟨e1, e2±e4, e5±ie3⟩ ⟨e1, e4±e2, e3±ie5⟩ , ⟨e1, e4±e2, e5±ie3⟩ α and β invariants for low-dimensional Lie algebras 85 Table 4.8: 6 -dimensional non-decomposable solvable non-nilpotent Lie algebras g Lie brackets α(g) Parameters ga,b,c,d 6,1 [e1, e3] = ae3,[e1, e4] = ce4,[e1, e6] = e6 [e2, e3] = be3,[e2, e4] = de4,[e2, e5] = e5 4ab = 0 c2+d2= 0 ga,b,c 6,2 [e1, e3] = ae3,[e1, e4] = e4,[e1, e5] = e6, [e2, e6] = e6,[e2, e3] = be3, [e2, e4] = ce4,[e2, e5] = e5 4a2+b2= 0 ga 6,3 [e1, e3] = e3,[e2, e5] = e5,[e1, e4] = e4, [e1, e5] = e6,[e2, e3] = ae3+e4, [e2, e4] = ae4,[e2, e6] = e6 4 ga,b 6,4 [e1, e3] = e3,[e1, e4] = e4,[e1, e5] = e6, [e2, e6] = ae6,[e2, e3] = e4, [e2, e4] = −e3,[e2, e5] = ae5+be6 4a= 0 ga,b 6,5 [e1, e3] = ae3,[e1, e5] = e5+e6,[e1, e6] = e6 [e2, e3] = be3,[e2, e4] = e4 4a= 0 ga,b 6,6 [e1, e3] = ae3,[e1, e4] = ae4,[e1, e5] = e5+e6 [e1, e6] = e6,[e2, e3] = e3+e4,[e2, e4] = e4 4a2+b2= 0 ga,b,c 6,7 [e1, e3] = ae3,[e1, e4] = ae4,[e2, e5] = be6, [e1, e6] = e6,[e2, e3] = ce3+e4, [e2, e4] = −e3+ce4,[e1, e5] = e5+e6 4a2+b2= 0 g6,8[e1, e3] = e3,[e1, e4] = e6,[e2, e4] = e4 [e2, e5] = e5+e6,[e2, e6] = e6 4 g6,9[e1, e3] = e3,[e1, e4] = e6,[e2, e6] = e6 [e2, e4] = e4+e5,[e2, e5] = e5+ae6 4 ga,b 6,10 [e1, e3] = ae3,[e1, e4] = e3+be6,[e1, e5] = e5 [e2, e5] = e6,[e1, e6] = e6, [e2, e3] = e3,[e2, e4] = e5 4 ga 6,11 [e1, e3] = e4,[e2, e3] = e3,[e2, e4] = e4, [e1, e6] = e6,[e2, e5] = ae5, [e1, e5] = e5+e6,[e2, e6] = ae6 4 ga,b 6,12 [e1, e3] = e3+e4,[e1, e5] = e5+e6 [e1, e4] = e4,[e1, e6] = e6, [e2, e3] = ae4+e5−be6,[e2, e4] = e6 [e2, e5] = −e3+be4+ae6,[e2, e6] == −e4 4 ga,b,c,d 6,13 [e1, e3] = ae3,[e1, e4] = ce4,[e1, e5] = e6, [e2, e5] = e5,[e1, e6] = −e5,[e2, e3] = be3, [e2, e4] = de4,[e2, e6] = e6 4 ga,b,c 6,14 [e1, e5] = ce5+e6,[e1, e6] = −e5+ce6 [e1, e3] = ae3,[e2, e3] = be3,[e2, e4] = e4 4ab = 0 ga,b,c,d 6,15 [e1, e4] = e4,[e1, e5] = ae5+be6 [e2, e6] = de6,[e1, e6] = −be5+ae6 [e2, e3] = ce3+e4,[e2, e5] = de5, [e1, e3] = e3,[e2, e4] = −e3+ce4 4b= 0 ga,b 6,16 [e1, e3] = e4,[e1, e5] = ae5+e6,[e2, e3] = e3, [e1, e6] = −e5+ae6,[e2, e4] = e4, [e2, e5] = be5,[e2, e6] = be6 4 ga 6,17 [e1, e3] = ae3+e4,[e1, e4] = ae4,[e1, e5] = e6 [e1, e6] = −e5,[e2, e5] = e5,[e2, e6] = e6 4 ga,b,c 6,18 [e1, e3] = e4,[e1, e5] = ae5+be6, [e1, e4] = −e3,[e2, e6] = de6, [e2, e4] = e4,[e1, e6] = −be5+ae6, [e2, e3] = e3,[e2, e5] = ce5 4b= 0 92 Chapter 4. Algorithm and applications Table 4.15: α invariant for nilpotent Lie algebras of dimension less than 6 . gdim(g) Lie brackets α(g) n33 [e1, e2] = e32 n44 [e1, e2] = e3,[e1, e3] = e43 g5,65 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e2, e3] = e53 g5,55 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e54 g5,35 [e1, e2] = e4,[e1, e4] = e5,[e2, e3] = e53 g5,45 [e1, e2] = e3,[e1, e3] = e4,[e2, e3] = e53 g5,25 [e1, e2] = e4,[e1, e3] = e54 g5,15 [e1, e3] = e5,[e2, e4] = e53 Table 4.16: α invariant for 6 -dimensional nilpotent Lie algebras Magnin de Graaf Seeley α(g) g6,20 L6,14 12346E3 g6,18 L6,16 12346C3 g6,19 L6,15 12346D4 g6,17 L6,17 12346B4 g6,15 L6,21(1) 1346C4 g6,13 L6,13 1246 4 g6,16 L6,18 12346A5 g6,14 L6,21(0) 2346 4 g6,9L6,19(1) 136A4 g6,12 L6,11 1346B4 g5,6⊕CL6,61 + 1235B4 g6,5L6,24(1) 246E4 g6,10 L6,20 136B4 g6,11 L6,12 1346A4 g5,5⊕CL6,71 + 1235A5 g6,8L6,24(0) 246D4 g6,4L6,19(0) 246B4 g6,7L6,23 246C4 g6,2L6,10 146 4 g6,6L6,25 246A5 g5,4⊕CL6,91 + 235 4 g5,3⊕CL6,51 + 135 4 n3⊕n3L6,22(1) 13 + 13 4 n4⊕C2L6,32 + 124 5 g6,1L6,22(0) 26 4 g6,3L6,26 36 4 g5,2⊕CL6,81 + 25 5 g5,1⊕CL6,41 + 15 4 n3⊕C3L6,23 + 13 5 C6L6,10 6 α and β invariants for low-dimensional Lie algebras 93 Table 4.17: 6 -dimensional non-decomposable nilpotent Lie algebras g Lie brackets g6,1[e1, e2] = e5,[e1, e4] = e6,[e2, e3] = e6. g6,2[e1, e2] = e5,[e1, e5] = e6,[e3, e4] = e6. g6,3[e1, e2] = e4,[e1, e3] = e5,[e2, e3] = e6. g6,4[e1, e2] = e4,[e1, e3] = e6,[e2, e4] = e5. g6,5[e1, e2] = e4,[e1, e4] = e5,[e2, e3] = e6,[e2, e4] = e6. g6,6[e1, e2] = e4,[e2, e3] = e6,[e2, e4] = e5. g6,7[e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e2, e3] = −e6. g6,8[e1, e2] = e4,[e1, e4] = e5,[e2, e3] = e5,[e2, e4] = e6. g6,9[e1, e2] = e4,[e1, e3] = e5,[e2, e5] = e6,[e3, e4] = e6. g6,10 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e3, e5] = e6. g6,11 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e2, e3] = e6. g6,12 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e2, e3] = e6,[e2, e4] = e6. g6,13 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e2, e3] = e5,[e3, e4] = −e6. g6,14 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e2, e3] = e6. g6,15 [e1, e2] = e3,[e1, e3] = e4,[e1, e5] = e6,[e2, e3] = e5,[e2, e4] = e6. g6,16 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6. g6,17 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6,[e2, e3] = e6. g6,18 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e2, e5] = e6,[e3, e4] = −e6. g6,19 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5, [e1, e5] = e6,[e2, e3] = e5,[e2, e4] = e6. g6,20 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5, [e2, e3] = e5,[e2, e5] = e6,[e3, e4] = −e6 . Table 4.18: 7 -dimensional non-decomposable nilpotent Lie algebras (I) g Lie brackets g7,0.1[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6,[e1, e6] = e7, [e2, e3] = e6,[e2, e4] = e7,[e2, e5] = e7,[e3, e4] = −e7 . g7,0.2[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6, [e1, e6] = e7,[e2, e3] = e5+e7,[e2, e4] = e6,[e2, e5] = e7 . g7,0.3[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6, [e1, e6] = e7,[e2, e3] = e6+e7,[e2, e4] = e7 . g7,0.4(λ) [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = λe7+e6,[e1, e5] = e7, [e1, e6] = e7,[e2, e3] = e5,[e2, e4] = e7,[e2, e5] = e6,[e3, e5] = e7 . g7,0.5[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e6+e7, [e1, e6] = e7,[e2, e3] = e5,[e2, e5] = e6,[e3, e5] = e7 . g7,0.6[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e7,[e1, e5] = e6,[e1, e6] = e7, [e2, e3] = e5,[e2, e4] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,0.7[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e7,[e1, e5] = e7,[e1, e6] = e7, [e2, e3] = e5,[e2, e4] = e7,[e2, e5] = e6,[e3, e5] = e7 . g7,0.8[e1, e2] = e4,[e1, e3] = e7,[e1, e4] = e5,[e1, e5] = e6, [e2, e3] = e6,[e2, e4] = e6,[e2, e6] = e7,[e4, e5] = −e7 . g7,1.01(i) [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e1, e6] = e7, [e2, e3] = e5+e7,[e3, e4] = −e6,[e3, e5] = −e7 . g7,1.01(ii) [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6, [e1, e6] = e7,[e2, e3] = e6+e7,[e3, e4] = −e7 . g7,1.02 [e1, e2] = e3,[e1, e3] = e4,[e1, e5] = e6,[e2, e3] = e5, [e2, e4] = e6,[e2, e5] = e7,[e2, e6] = e7,[e3, e5] = −e7 . 94 Chapter 4. Algorithm and applications Table 4.19: 7 -dimensional non-decomposable nilpotent Lie algebras (II) g Lie brackets g7,1.03 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e6] = e7, [e2, e3] = e6,[e2, e4] = e7,[e2, e5] = e7,[e3, e4] = −e7 . g7,1.1(iλ) [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5, [e1, e5] = e6,[e1, e6] = e7,[e2, e3] = e5, [e2, e4] = e6,[e2, e5] = λe7,[e3, e4] = e7(−λ+ 1) . g7,1.1(ii) [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6, [e1, e6] = e7,[e2, e5] = e7,[e3, e4] = −e7 . g7,1.1(iii) [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6, [e2, e3] = e5,[e2, e4] = e6,[e2, e5] = −e7,[e3, e4] = e7 . g7,1.1(iv) [e1, e2] = e3,[e1, e3] = e4,[e1, e5] = e6,[e1, e6] = e7, [e2, e3] = e5,[e2, e4] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,1.1(v) [e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6,[e1, e6] = e7, [e2, e3] = e5,[e2, e4] = e6,[e2, e5] = e7,[e3, e4] = −e7 . g7,1.1(vi) [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e6] = e7, [e2, e3] = e5,[e2, e5] = e7,[e3, e4] = −e7 . g7,1.2(iλ) [e1, e2] = e4,[e1, e3] = e6,[e1, e4] = e5,[e1, e5] = e7, [e2, e3] = λe5,[e2, e4] = e6,[e2, e6] = e7,[e3, e4] = e7(−λ+ 1) . g7,1.2(ii) [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e7,[e1, e6] = e7 [e2, e3] = e6,[e2, e4] = e6,[e2, e5] = e7,[e3, e4] = −e7 . g7,1.2(iii) [e1, e2] = e4,[e1, e3] = e6,[e1, e4] = e5,[e1, e5] = e7, [e2, e3] = e5,[e2, e4] = e6,[e3, e4] = −e7 . g7,1.2(iv) [e1, e2] = e4,[e1, e4] = e6,[e1, e5] = −e7,[e1, e6] = e7, [e2, e3] = e5,[e2, e5] = e7,[e3, e4] = e7 . g7,1.3(iλ) [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e1, e6] = e7, [e2, e3] = e6,[e2, e4] = λe7,[e2, e5] = e7,[e3, e5] = e7 . g7,1.3(ii) [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e1, e6] = e7, [e2, e3] = e6,[e2, e4] = e7,[e2, e5] = e7 2,[e3, e4] = −e7 2 . g7,1.3(iii) [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6, [e1, e6] = e7,[e2, e4] = e7,[e3, e5] = e7 . g7,1.3(iv) [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6, [e2, e3] = e6,[e2, e4] = e7,[e3, e5] = e7 . g7,1.3(v) [e1, e2] = e4,[e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e6 [e2, e4] = e7,[e3, e4] = −e7,[e3, e5] = −e7 . g7,1.4[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6 [e1, e6] = e7,[e2, e3] = e6,[e2, e4] = e7 . g7,1.5[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6 [e2, e3] = e6,[e2, e5] = −e7,[e3, e4] = e7 . g7,1.6[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5, [e1, e5] = e6,[e1, e6] = e7,[e2, e3] = e7 . g7,1.7[e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6, [e1, e5] = e7,[e2, e3] = e6,[e2, e4] = e7 . g7,1.8[e1, e2] = e4,[e1, e4] = e6,[e1, e6] = e7, [e2, e3] = e5,[e2, e4] = e7,[e3, e5] = e7 . g7,1.9[e1, e2] = e4,[e1, e3] = e6,[e1, e4] = e5, [e1, e5] = e7,[e2, e3] = e7,[e2, e4] = e6 . g7,1.10 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e6, [e1, e6] = e7,[e2, e3] = e5,[e2, e5] = e7 . g7,1.11 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e1, e6] = e7 [e2, e3] = e6,[e2, e4] = e6,[e2, e5] = e7,[e3, e4] = −e7 . g7,1.12 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e1, e6] = e7 [e2, e3] = e7,[e2, e4] = e6,[e2, e5] = e7 . g7,1.13 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e6,[e1, e5] = e7 [e1, e6] = e7,[e2, e3] = e5,[e2, e4] = e7 . g7,1.14 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e6,[e2, e3] = e5 [e2, e5] = −e7,[e2, e6] = −e7,[e3, e4] = e7 . α and β invariants for low-dimensional Lie algebras 95 Table 4.20: 7 -dimensional non-decomposable nilpotent Lie algebras (III) g Lie brackets g7,1.15 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6, [e1, e6] = e7,[e2, e3] = e7,[e2, e4] = e7 . g7,1.16 [e1, e2] = e3,[e1, e3] = e5,[e1, e4] = e6, [e1, e5] = e7,[e1, e6] = e7,[e2, e3] = e7 . g7,1.17 [e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e5, [e2, e5] = e6,[e2, e6] = e7,[e3, e4] = −e7,[e3, e5] = e7 . g7,1.18 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e7, [e2, e3] = e6+e7,[e2, e4] = e6 . g7,1.19 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e7, [e1, e5] = e6,[e2, e4] = e6,[e3, e5] = e7 . g7,1.20 [e1, e2] = e3,[e1, e5] = e6,[e1, e6] = e7,[e2, e3] = e4, [e2, e4] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,1.21 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e2, e3] = e6, [e2, e4] = e6,[e2, e6] = e7,[e4, e5] = −e7 . g7,2.1(iλ) [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e1, e6] = e7, [e2, e3] = e6,[e2, e5] = λe7,[e3, e4] = e7(λ−1) . g7,2.1(ii) [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6, [e1, e6] = e7,[e2, e5] = e7,[e3, e4] = e7 . g7,2.1(iii) [e1, e2] = e4,[e1, e4] = e6,[e1, e6] = e7, [e2, e3] = e6,[e2, e5] = e7,[e3, e4] = −e7 . g7,2.1(iv) [e1, e3] = e5,[e1, e4] = e6,[e1, e6] = e7, [e2, e3] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,2.1(v) [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6, [e2, e3] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,2.2[e1, e2] = e5,[e1, e3] = e6,[e1, e4] = 2e7,[e2, e3] = e4, [e2, e6] = e7,[e3, e5] = −e7,[e3, e6] = e7 . g7,2.3[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5,[e1, e5] = e6,[e1, e6] = e7 . g7,2.4[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e5, [e1, e5] = e6,[e2, e5] = −e7,[e3, e4] = e7 . g7,2.5[e1, e2] = e3,[e1, e3] = e4,[e1, e5] = e6,[e1, e6] = e7, [e2, e3] = e5,[e2, e4] = e6,[e3, e4] = e7 . g7,2.6[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e6, [e2, e3] = e5,[e2, e6] = e7,[e3, e4] = −e7 . g7,2.7[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e5 . g7,2.8[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e6,[e1, e5] = e7,[e2, e3] = e5,[e2, e4] = e7 . g7,2.9[e1, e2] = e3,[e1, e3] = e4,[e1, e4] = e6,[e2, e3] = e5,[e2, e5] = e7 . g7,2.10 [e1, e2] = e4,[e1, e3] = e7,[e1, e4] = e5,[e1, e5] = e6,[e2, e6] = e7,[e4, e5] = −e7 . g7,2.11 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e7,[e1, e6] = e7,[e2, e3] = e6,[e2, e4] = e6 . g7,2.12 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e2, e4] = e7,[e3, e5] = e7 . g7,2.13 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e2, e3] = e6,[e2, e6] = e7,[e4, e5] = −e7 . g7,2.14 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e1, e6] = e7, [e2, e3] = e5,[e3, e4] = −e6,[e3, e5] = −e7 . g7,2.15 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6, [e1, e6] = e7,[e2, e3] = e6,[e3, e4] = −e7 . g7,2.16 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6,[e1, e6] = e7,[e2, e3] = e7 . g7,2.17 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e7,[e2, e3] = e5,[e2, e4] = e6,[e3, e4] = −e7 . g7,2.18 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e7,[e2, e3] = e7,[e2, e4] = e6 . g7,2.19 [e1, e2] = e4,[e1, e3] = e6,[e1, e4] = e5,[e1, e5] = e7,[e2, e4] = e6 . g7,2.20 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e6,[e1, e6] = e7,[e3, e5] = e7 . g7,2.21 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e7 . g7,2.22 [e1, e3] = e4,[e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e5,[e3, e4] = e7 . 96 Chapter 4. Algorithm and applications Table 4.21: 7 -dimensional non-decomposable nilpotent Lie algebras (IV) g Lie brackets g7,2.23 [e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e5,[e2, e5] = e7,[e3, e4] = e7 . g7,2.24 [e1, e2] = e4,[e1, e4] = e6,[e1, e5] = −e7,[e1, e6] = e7,[e2, e3] = e5,[e3, e4] = e7 . g7,2.25 [e1, e2] = e5,[e1, e5] = e6,[e1, e6] = e7,[e2, e3] = e6,[e3, e4] = −e7,[e3, e5] = −e7 . g7,2.26 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e6,[e2, e5] = e7,[e3, e4] = e7,[e3, e5] = e6 . g7,2.27 [e1, e2] = e5,[e1, e3] = e7,[e1, e5] = e6,[e2, e4] = e7,[e2, e5] = e7 . g7,2.28 [e1, e2] = e5,[e1, e3] = e6,[e1, e6] = e7,[e2, e5] = e7,[e3, e4] = e7 . g7,2.29 [e1, e2] = e5,[e1, e5] = e6,[e2, e3] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,2.30 [e1, e2] = e5,[e1, e5] = e6,[e1, e6] = e7,[e2, e5] = e7,[e3, e4] = e7 . g7,2.31 [e1, e2] = e4,[e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e5,[e2, e5] = e7 . g7,2.32 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e1, e6] = e7,[e3, e5] = e7 . g7,2.33 [e1, e2] = e4,[e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e5,[e3, e5] = e7 . g7,2.34 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e7,[e2, e4] = e6,[e3, e5] = e7 . g7,2.35 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e6,[e2, e4] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,2.36 [e1, e3] = e5,[e1, e4] = e7,[e2, e3] = e6,[e2, e4] = −e5,[e3, e6] = −e7 . g7,2.37 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e5,[e1, e6] = e7, [e2, e4] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,2.38 [e1, e2] = e3,[e1, e3] = e6,[e1, e5] = e7,[e2, e3] = e7,[e2, e4] = e6 . g7,2.39 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e1, e5] = e7,[e2, e3] = e6 . g7,2.40 [e1, e2] = e3,[e1, e3] = e5,[e1, e4] = e7,[e2, e3] = e6,[e2, e4] = e5 . g7,2.41 [e1, e2] = e3,[e1, e3] = e5,[e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e6,[e2, e5] = e7 . g7,2.42 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e6,[e2, e3] = e6,[e2, e4] = e7 . g7,2.43 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e7,[e1, e5] = e6,[e3, e5] = e7 . g7,2.44 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e7,[e2, e3] = e6,[e2, e4] = e7 . g7,2.45 [e1, e2] = e5,[e1, e4] = e7,[e1, e5] = e6,[e2, e3] = e7,[e2, e4] = e6 . g7,3.1(iλ) [e1, e2] = e4,[e1, e3] = e5,[e1, e6] = e7, [e2, e3] = e6,[e2, e5] = λe7,[e3, e4] = e7(λ−1) . g7,3.1(iii)[e1, e2] = e4,[e1, e3] = e5,[e1, e6] = e7,[e2, e5] = e7,[e3, e4] = e7 . g7,3.2[e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e1, e6] = e7 . g7,3.3[e1, e2] = e4,[e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e5 . g7,3.4[e1, e2] = e4,[e1, e3] = e5,[e2, e4] = e6,[e3, e5] = e7 g7,3.5[e1, e2] = e4,[e1, e3] = e5,[e2, e4] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,3.6[e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e7,[e2, e3] = e6 . g7,3.7[e1, e2] = e5,[e1, e5] = e6,[e2, e4] = e6,[e3, e4] = −e7 . g7,3.8[e1, e2] = e5,[e1, e3] = e6,[e1, e5] = e7,[e2, e4] = e7 . g7,3.9[e1, e2] = e5,[e1, e5] = −e7,[e2, e3] = e6,[e2, e4] = e7 . g7,3.10 [e1, e2] = e5,[e1, e3] = e6,[e2, e4] = e6,[e2, e6] = e7,[e3, e5] = e7 . g7,3.11 [e1, e2] = e5,[e1, e3] = e6,[e1, e5] = e7,[e2, e4] = e6 . g7,3.12 [e1, e2] = e5,[e1, e3] = e6,[e2, e4] = e6,[e3, e4] = e7 . g7,3.13 [e1, e2] = e5,[e1, e5] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,3.14 [e1, e2] = e5,[e1, e3] = e6,[e1, e6] = e7,[e2, e4] = e7 . g7,3.15 [e1, e2] = e5,[e1, e3] = e6,[e2, e5] = e7,[e3, e4] = e7 . g7,3.16 [e1, e2] = e5,[e1, e5] = e7,[e3, e4] = e6,[e3, e6] = e7 . g7,3.17 [e1, e2] = e5,[e1, e5] = e6,[e1, e6] = e7,[e3, e4] = e7 . g7,3.18 [e1, e2] = e6,[e1, e6] = e7,[e2, e5] = e7,[e3, e4] = e7 . g7,3.19 [e1, e2] = e6,[e1, e3] = e7,[e3, e4] = e6,[e4, e5] = e7 . g7,3.20 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e1, e5] = e7 . g7,3.21 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6,[e3, e5] = e7 . g7,3.22 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e7,[e2, e5] = e6,[e3, e4] = e6 . g7,3.23 [e1, e2] = e3,[e1, e3] = e5,[e1, e4] = e7,[e2, e3] = e6 . g7,3.24 [e1, e2] = e5,[e2, e3] = e6,[e2, e4] = e7,[e3, e4] = e5 . g7,4.1[e1, e2] = e5,[e1, e3] = e6,[e3, e4] = e7 . g7,4.2[e1, e2] = e5,[e1, e3] = e6,[e1, e4] = e7 . g7,4.3[e1, e2] = e6,[e3, e5] = e6,[e4, e5] = e7 . g7,4.4[e1, e4] = e7,[e2, e5] = e7,[e3, e6] = e7 . Computation of α invariant for low-dimensional Lie algebras 97 4.3 Matrix representation of liform Lie algebras In this section we obtain a minimal faithful unitriangular matrix representation for each model liform Lie algebra. Additionally, we also introduce a method for obtaining such representations for non-model liform Lie algebras, debugging and improving the one given in [7]. In order to compute those representations, we have used the Lie algebras gn of n×n strictly upper-triangular matrices. Therefore, we have taken advantage of the previous calculations done in Chapter 3 to compare the abelian subalgebras and ideals in Lie algebras gn and the unique abelian ideal of maximal dimension for each liform Lie algebra, as it was proved in Proposition 3.2. This can be seen in Example 4.3 and in Tables 4.22-4.24 where we have computed the minimal faithful unitriangular matrix representations of liform Lie algebras with dimension less than 9 . Given a Lie algebra g , a representation of g in Cn is a homomorphism of Lie algebras ϕ:g→gl(Cn) = gl(C, n) . The natural integer n is called the dimension of this representation. Ado's theorem states that every nite-dimensional Lie algebra over a eld of characteristic zero (as in the case of C ) has a linear injective representation on a nite-dimensional vector space, that is, a faithful representation . Usually, representations are dened by using an arbitrary n -dimensional vector space V (like in [43]) and homomorphisms of Lie algebras from g to gl(V) of endomorphisms of V ; that is, by using g -modules . With respect to minimal representations of Lie algebras, Burde [15] introduced the following invariant value for an arbitrary Lie algebra g µ(g) = min{dim(M)|M is a faithful g -module }. The invariant µ(g) plays an important role in the theory of ane at manifolds and ane crystallographic groups, see [17]. Moreover, there exist several relations between this invariant and α invariant of Lie algebras. For example, if g is a Lie algebra with dim(D(g)) = 1 , then µ(g)> dim(g)−α(g) + 1 . In this section, matrix representations of liform Lie algebras are studied. Moreover, we are interested in minimal matrix representations of these algebras with a particular restriction: the representations have to be contained in gn . In this way, given a liform Lie algebra g , we want to compute the minimal value n such that gn contains a subalgebra isomorphic to g . This value is also an invariant of g and its expression is given by ¯µ(g) = min{n∈N| ∃ subalgebra of gn isomorphic to g}. Let us note that the invariants µ(g) and ¯µ(g) can be dierent from each other. 98 Chapter 4. Algorithm and applications Proposition 4.7. Let g be an n -dimensional liform Lie algebra. Then ¯µ(g)≥n . Proof. For a given n -dimensional liform Lie algebra g , we have to prove that it is not possible to nd a subalgebra of gn−1 isomorphic to g . First, we write the vectors of an adapted basis {ei}n i=1 of g as linear combinations of the vectors in the basis Bn−1 of gn−1 ek=∑ 1≤i<j≤n−1 λk i,jXi,j,for 1 ≤k≤n. We will prove that all the coecients λ2 i,j of e2∈Z(g) have to be zero. From the equation [e1, eh] = eh−1 , for 3≤h≤n , the following relations are obtained λh−1 β,β+1 = 0, λh−1 β,αβ=∑β<p<αβ(λ1 β,pλh p,αβ−λ1 p,αβλh β,p),}for {1≤β≤n−2; αβ≥β+ 2. (4.1) From the equality [e1, e3] = e2 , we can conclude that λ2 β,β+1 = 0 , for 1≤β≤ n−2 . Now, we have to prove that λ2 l,αl= 0 , for 1≤l≤n−3 . To do so, we are going to prove that λ3 p,αβ=λ3 β,p = 0 in each case. From the equality [e1, ek] = ek−1 , for 3≤k≤n−1 , we can arm that λk−1 β,β+1 = 0 , for 1≤β≤n−2 . This implies that λ3 p,q = 0 , when q−p < n −4 . If we consider the bracket [e1, en] = en−1 , we conclude that λn−1 β,β+1 = 0 , and, therefore, λ3 p,q = 0 , where q−p=n−3 . Consequently, all the coecients of e2 are null and this is a contradiction. 4.3.1 Model liform Lie algebras The law of a xed n -dimensional model liform Lie algebra g with an adapted basis {ei}n i=1 is the following [e1, eh] = eh−1,for 3 ≤h≤n. (4.2) Now, we will construct the n -dimensional subalgebra f′ n of gn and its law is exactly the same of the model liform Lie algebra g . Just dene a basis {ei}n i=1 of this subalgebra as linear combinations of the vectors in the basis Bn of the Lie algebra gn e1= n−2 ∑ i=1 Xi,i+1, e2=X1,n, e3=X2,n, . . . , en=Xn−1,n (4.3) Matrix representation of liform Lie algebras 99 Consequently, we have dened the subalgebra f′ n whose elements have the following form f′ n(xk) =            0x10··· 0x2 0 0 x1··· 0x3 . . .. . .. . ..... . .. . . 0 0 0 ··· x1xn−1 0 0 0 ··· 0xn 0 0 0 ··· 0 0            (xk∈C,for k= 1, . . . , n). With respect to this faithful matrix representation, the adapted basis {ei}n i=1 of f′ n is given as follows eh=f′ n(xk),with xk={1,if k=h; 0,if k=h. According to the reasoning just given above and Proposition 2 , we can arm the following result Proposition 4.8. Every n -dimensional model liform Lie algebra has an n -dimen- sional minimal faithful unitriangular matrix representation. Moreover, a representative is given by the Lie algebra f′ n given in (4.3).  4.3.2 Non-model liform Lie algebras If the liform Lie algebra g is non-model, then the invariants z1 and z2 exist. Hence, there exist some additional non-zero brackets to [e1, eh] = eh−1 , for 3≤h≤n . Consequently, non-model liform Lie algebras cannot be represented by Lie algebras f′ n . Now, we show an algorithmic method to compute minimal faithful unitriangular matrix representations for non-model liform Lie algebras. These representations are minimal in the following sense: nding a faithful matrix representation of a given Lie algebra g in gn , but no representations of g can be obtained in gn−1 . To do so, we give a step-by-step explanation of the method used to determine these minimal representations for a given liform Lie algebra g of dimension n > 4 . Let us note that this algorithmic method debugs and improves the one introduced previously in [7] to be applied for any liform Lie algebra, not necessarily model. When imposing the condition of being non-model, remarkable improvements have been achieved in its implementation with respect to computing time and memory usage. 100 Chapter 4. Algorithm and applications 1. According to Proposition 1.1, we have to compute the rst natural integer l such that the lower central series of gl is compatible with the one associated with g . By bearing in mind Proposition 4.7, we can start considering l=n , for n -dimensional liform Lie algebras. Hence, we have ruled out the Lie algebra gl with l < n . 2. Now, we search a subalgebra of gl isomorphic to g , where l≥n and l is as small as posible. To do so, an adapted basis {ei}n i=1 of g is considered and its vectors are expressed as linear combinations of the basis Bl eh=∑ 1≤i<j≤l λh i,jXi,j,for 1 ≤h≤n. 3. The next step is to impose the brackets given in (4.2), obtaining again the equations shown in (4.1), but with respect to the algebra gl considered by the method. 4. After solving the system of equations resulting from the previous step, we solve a new system obtained by imposing the rest of the brackets in the law of g . In particular, we will compare the abelian subalgebras of maximal dimension of g and gl . Obviously, the solutions of this system depend on the particular Lie algebra studied in each moment. Hence, we have generalized some cases by using invariants z1 and z2 in Section 4.3.4. The solutions of the system given in Step 4 have been computed by using the command solve of the symbolic computation package Maple 12. This command works eciently with polynomial equations, receives as inputs the list of equations and the list of variables, and returns as output the algebraic expression of the solutions. After checking the existence of representations in the Lie algebra gl , we search a natural representative in the sense of considering the following conditions: e2∈ ⟨X1,l⟩ and there exist the greatest possible number of null-parameters. The rst condition is due to the fact that Z(gl) = ⟨X1,l⟩ and is also in concordance with Proposition 2 . Another point to consider is the number of solutions for the system. This number can be studied by dening the set F of polynomial expressions and by using the command is finite , which determines if the number of solutions is or not nite for the system dened by the input set F . Likewise, the Noether normalization lemma is also very useful to describe the elements in an algebraic variety. Matrix representation of liform Lie algebras 101 Furthermore, in order to compute a particular solution of the previous system, we have searched one such that the number of null coecients is as greater as possible. In this way, the coecients could be assumed equal to zero when they do not appear in the relations obtained. This will be a natural representative of the Lie algebra g . 4.3.3 Examples of application Next, we give two examples of computing minimal faithful matrix representations for liform Lie algebras. The rst is referred to a model liform Lie algebra and, afterwards, we apply our algorithmic method (given in the previous section) to compute this type of representation for a non-model liform Lie algebra. Example 4.1. Consider model liform Lie algebra f1 4 generated by {ei}4 i=1 with law [e1, e3] = e2,[e1, e4] = e3 . We can compute its minimal faithful unitriangular matrix representation as was shown in Subsection 4.3.1 by using the Lie algebra f′ 4 . However, although we can obtain directly a representative of such representation, we might be interested in determining how many representations the algebra f1 4 has: a nite or innite amount. To do so, it is necessary to apply the algorithm given for non-model liform Lie algebras in Subsection 4.3.2, but directly over its dimension, 4, and only using Steps 3 and 4. In this way, we determine if there is a nite or innite number of solution of the system obtained. First, we dene the vectors e1=λ1 1,2X1,2+λ1 1,3X1,3+. . . +λ1 3,4X3,4, e2=λ2 1,2X1,2+λ2 1,3X1,3+. . . +λ2 3,4X3,4, e3=λ3 1,2X1,2+λ3 1,3X1,3+. . . +λ3 3,4X3,4, e4=λ4 1,2X1,2+λ4 1,3X1,3+. . . +λ4 3,4X3,4. Now, we introduce the following commands in MAPLE, taking into account that λk i,j is denoted by cijk 108 Chapter 4. Algorithm and applications Table 4.23: Minimal faithful matrix representations for dimension 8 (I) Law Representation f1 8 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X3,8, e5=X4,8, e6=X5,8, e7=X6,8, e8=X7,8 . f2 8[e7, e8] = e2 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X3,8, e5=X4,8, e6=X5,8, e7=X1,7+X6,8, e8=X2,7+X7,8 . f3 8 [e6, e8] = e2 , [e7, e8] = e3 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X3,8, e5=X4,8, e6=X5,8, e7=X6,8, e8=−X1,5−X2,6−X3,7+X7,8 . f4 8 [e6, e8] = e2 , [e7, e8]= e2+e3 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X3,8, e5=X4,8, e6=X5,8, e7=X6,8, e8=−X1,5−X1,6−X2,6−X2,7−X3,7+X7,8 . f5 8 [e5, e8] = e2 , [e6, e8] = e3 [e7, e8]= e2+e4 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X3,8, e5=X4,8, e6=X5,8, e7=X6,8, e8=−X1,4−X1,6−X2,5−X2,7−X3,6−X4,7+X7,8 f6 8 [e5, e8] = e2 , [e6, e8]= e2+e3 [e7, e8]= αe2+e3+e4 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X3,8, e5=X4,8, e6=X5,8, e7=X6,8, e8=−X1,4−X1,5−αX1,6+X1,8 − X2,5−X2,6−αX2,7−X3,6−X3,7−X4,7+X7,8 f7 8 [e5, e8] = αe2 , [e6, e7] = e2 [e6, e8]= (1+α)e3 [e7, e8]= (1+α)e4 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X3,8, e5=−X1,7+X4,8, e6=X1,6+X5,8, e7=X2,6+X3,7+X6,8, e8=−(1+α)X1,4+X1,8 −(1+α)X2,5−αX3,6+(1−α)X4,7+X7,8 f8 8 [e5, e8] = αe2 , [e6, e7] = e2 [e6, e8]=(1+α)e3+e2 [e7, e8]=(1+α)e4+e3 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X3,8, e5=−X1,7+X4,8, e6=X1,6+X5,8, e7=X2,6+X3,7+X6,8, e8=−(1+α)X1,4−X1,5+X1,8−(1+α)X2,5−X2,6 −αX3,6−X3,7+(1−α)X4,7+X7,8. f9 8 [e4, e8] = e2 , [e5, e8] = e3 [e6, e8] = e4 , [e7, e8] = e5 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8 , e4=X1,7+X3,8, e5=X2,7+X4,8, e6=−1 2X1,5−1 2X2,6+1 2X3,7+X5,8, e7=−1 2X2,5−X3,6−1 2X4,7+X6,8, e8=−1 2X3,5−3 2X4,6−2X5,7+X7,8 . f10 8 [e4, e8] = e2 , [e5, e8] = e3 [e6, e8] = e4 , [e7, e8]= e2+e5 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X1,7+X3,8, e5=X2,7+X4,8, e6=−1 2X1,5−1 2X2,6+1 2X3,7+X5,8, e7=−1 2X2,5−X3,6−1 2X4,7+X6,8, e8=−X1,6−X2,7−1 2X3,5−3 2X4,6−2X5,7+X7,8 . f11 8 [e4, e8] = e2 , [e5, e8] = e3 [e6, e8]= e2+e4, [e7, e8]= αe2+e3+e5 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X1,7+X3,8, e5=X2,7+X4,8, e6=−1 2X1,5−1 2X2,6+1 2X3,7+X5,8, e7=−1 2X2,5−X3,6−1 2X4,7+X6,8, e8=−X1,5−αX1,6 −X2,6−αX2,7−1 2X3,5−X3,7−3 2X4,6−2X5,7+X7,8 . f12 8 [e4, e8] = e2 , [e5, e8] = e3 , [e6, e7] = e2, [e6, e8]= αe2+e3+e4 [e7, e8]= αe3+e4+e5 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X1,7+X3,8, e5=−2X1,7+X2,7+X4,8, e6=−1 2X1,5+X1,6−1 2X2,6 −X2,7+1 2X3,7+X5,8, e7=−1 2X2,5+X2,6−X3,6 − 1 2X4,7+X6,8, e8=−2X1,4−αX1,5−2X2,5−αX2,6 −1 2X3,5−X3,6−αX3,7−3 2X4,6−X4,7−2X5,7+X7,8 . f13 8 [e4, e8] = −e2 , [e6, e8] = e3+e4, [e6, e7] = e2+e3 , [e5, e7] = e2, [e7, e8] = e4+e5 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=X3,8, e5=X1,6+X2,7+X4,8, e6=−X1,5+1 2X1,6+1 2X2,7+X3,7+X5,8, e7=−1 2X1,5−X2,5−X3,6+1 2X3,7+X6,8, e8=X1,3+X2,4−1 2X2,5−1 2X3,6−X4,6−X5,7+X7,8. f14 8 [e4, e8] = αe2 , [e5, e7] = e2 [e5, e8] = −e3 , [e6, e7] = e3 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=−2X1,7+X3,8, e5=X1,6−X2,7+X4,8, e6=X2,6+X5,8, e7=X3,6+X4,7+X6,8, e8=X4,6+2X5,7+X7,8 . Matrix representation of liform Lie algebras 109 Table 4.24: Minimal faithful matrix representations for dimension 8 (II) Law Representation f15 8 [e4, e8]=−2e2 , [e5, e7]=e2 [e5, e8]=−e3+e2 , [e6, e7]=e3 [e6, e8]=e3, [e7, e8]=e4 e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=X2,8, e4=−2X1,7+X3,8, e5=X1,6+3X1,7− X2,7+X4,8, e6=X2,6+3X2,7+X5,8, e7=X3,6+X3,7+X4,7+X6,8, e8=2X1,4+2X2,5+2X3,6+ X4,6+ 5X4,7+2X5,7+X7,8 . f16 8 [e4, e7] = e2 , [e5, e6] = −e2 [e4, e8] = e3 [e5, e8] = e4, [e6, e8] = e5 , [e7, e8] = e6. e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=−1 2X1,7+ X2,8, e4=1 2X1,6+X3,8, e5=−1 2X1,5+ X4,8, e6=1 2X1,4+X5,8, e7=−1 2X1,3 +X6,8, e8=−1 2X1,2−X2,3−X3,4−X4,5 −X5,6−X6,7+X7,8 . f17 8 [e4, e7] = e2 , [e4, e8] = e3 [e5, e6] = −e2 , [e5, e8] = e4, [e6, e7] = e2 , [e6, e8] = e3+e5 [e7, e8] = e4+e6. e1=∑6 i=1 Xi,i+1, e2=X1,8, e3=−1 2X1,7 +X2,8, e4=1 2X1,6+X3,8, e5=−1 2X1,5− 3X1,7+X4,8, e6=1 2X1,4+X1,6 −2X2,7+X5,8, e7=−1 2X1,3+X2,6−X3,7 +X6,8, e8=−3X1,4−1 2X1,2−X2,3− 3X2,5−X3,4−2X3,6−X4,5− 3X4,7−X5,6−X6,7+X7,8 . f18 8 [e4, e7] = e2 , [e4, e8] = e3 [e5, e6] = −e2 , [e5, e8] = e4, [e6, e8] = e2+e5 [e7, e8] = e3+e6. e1=∑6 i=1 Xi,i+1 +3 40 X7,8, e2=X1,8, e3=−10 3X1,7+3 4X2,8, e4=4 3X1,6−2X2,7 +3 5X3,8, e5=−4 3X1,5−2X3,7+9 20 X4,8, e6=4 3X1,4−2X4,7+3 10 X5,8, e7=−4 3X1,3 −2X5,7+3 20 X6,8, e8=1 3X1,2 −5X1,5−X2,3−5X2,6−X3,4−5X3,7 −X4,5−3 8X4,8−X5,6−3X6,7−3 40 X7,8 . f19 8 [e4, e7]=e2,[e4, e8]=e3 [e5, e6]=−e2,[e5, e8]=e4, [e6, e7] = e2, [e6, e8] = e2+e3+e5 [e7, e8] = e3+e4+e6. e1=X1,2+X1,3−X2,3+ 164X2,5−X3,4− 164X3,6+164 3X3,7−164X4,5+ 164X4,7 −X5,6−X6,7−1 328 X7,8, e2=X1,8, e3= 164X1,7+1 2X2,8, e4= 164X1,6+ 164X1,7− 1 2X3,8, e5= 164X1,5+ 164X1,6−X1,8+ 1 2X4,8, e6=X1,4+ 164X1,5+ 164X1,6− 164 3X1,7−1 3X2,8−1 2X3,8−1 328 X5,8, e7=X1,3+X1,4+820 3X1,6−328 3X2,7 +1 328 X6,8, e8=X2,3−164 3X2,6+ X3,4+ 328X3,6+ 164X4,5+X5,6+X6,7 . f20 8 [e4, e7]=e2,[e4, e8]=e2+e3 [e5, e6]=−e2,[e5, e7]=−2 5e2, [e5, e8] = e4+3 5e3, [e6, e7] = −2 5e3, [e6, e8] = e5+1 5e4, [e7, e8] = e6+1 5e5. e1=X1,2+X1,3−X2,3+3 5X2,4−37 25 X2,5 −X3,4+2 5X3,5−2 5X3,6+1 25 X3,7 −3 250 X3,8−X4,5−2 5X4,6−2 25 X4,7 −X5,6−3 5X5,7−X6,7−1 2X7,8, e2=X1,8, e3=X1,7−X1,8+1 2X2,8, e4=X1,6+3 5X1,8−1 2X2,8−1 2X3,8, e5=X1,5+3 5X1,6+1 5X1,7−3 25 X1,8 −6 25 X2,7+3 10 X2,8+1 5X3,8+1 2X4,8, e6=X1,4+4 5X1,5+1 5X1,6+6 25 X1,7+ 3 125 X1,8−3 5X2,6−29 25 X2,7−1 25 X2,8+ 3 5X3,7−1 5X3,8−1 10 X4,8−1 2X5,8 , e7=X1,3+X1,4+7 25 X1,6+1 5X2,5− 7 5X2,6−2 25 X2,7+3 250 X2,8+4 5X3,6+ 1 5X4,7+1 2X6,8, e8=X2,3−1 25 X2,6 +X3,4+8 25 X3,6+X4,5+X5,6+X6,7 . Conclusions In this dissertation, we have dealt with abelian subalgebras and ideals of maximal dimension for arbitrary Lie algebras and, as a particular case, for solvable Lie algebras. We have also analyzed several special families of solvable Lie algebras in relation to this notion. We have studied this from both a theoretical view-point and an algorithmic approach. In this way, Chapter 2 constitutes a theoretical study of abelian subalgebras and ideals contained in Lie algebras. We have given several general properties and bounds for α and β invariants in cases of supersolvable, solvable and nilpotent Lie algebras. Then, we have focused on the cases of Lie algebras containing abelian subalgebras of codimension 1 and 2 , and the case of codimension 3 for nilpotent Lie algebras. In Chapter 3 , we have shown algorithmic methods to compute abelian subalgebras and ideals of maximal dimension for the most important families of solvable Lie algebras: Lie algebra gn , Lie algebra hn and Heisenberg algebra Hk . In order to conclude the chapter, we have studied liform Lie algebras, proving that there exists a unique abelian ideal of maximal dimension for these algebras. In fact,we do not only have analyzed α and β invariants with respect to the dimension of a given Lie algebra, but we have also obtained a relation allowing us to determine the law of a general liform Lie algebra. At this respect, we have given a characterization of the law of a general liform Lie algebra and we have proved several result about the coecients in those laws. In Chapter 4 , we have shown an algorithmic method to compute abelian subalgebras and ideals of any nite-dimensional Lie algebra, starting from the non-zero brackets in its law. Moreover, we included in this chapter two dierent applications of the previous results. The rst was related to the computation of α and β invariants for low dimensional Lie algebras. The second application was the computation of minimal faithful unitriangular matrix representations of liform Lie algebras and the use of these invariants to retrieve these representations. In our opinion, the results obtained in this dissertation can be considered a new step forward in the study of Lie algebras in general. More concretely, we believe 111 112 Conclusions that our main contribution is related to the study of abelian subalgebras and ideals of maximal dimension in Lie algebras and more concretely on solvable ones. We have also determined laws for liform Lie algebras starting from the value of the maximal dimension of their abelian ideals. Therefore, we think that this can be considered another step towards the problem of classifying liform Lie algebras in general, which seems to be a very dicult task. Moreover, we have also given several algorithmic methods that we believe that can be very useful to compute abelian subalgebras and ideals in Lie algebras, as well as making easier the computation of matrix representations of liform Lie algebras. In fact, besides explaining and implementing algorithms to compute the value of α and β invariants for any given Lie algebra, we have also applied them to several types of Lie algebras; namely: general, solvable and nilpotent for dimension less than 5 , 7 and 8 , respectively. Finally, we have also determined a representative for the matrix representation of general model liform Lie algebras and, applying our general algorithmic method, for all liform Lie algebras up to dimension 8 . However, there exist several open problems that we want to study in the near future. Regarding the second chapter, we have the following questions 1. Does Theorem 2.2 hold for supersolvable Lie algebras? 2. Let g be a supersolvable/nilpotent Lie algebra with α(g) = n−k containing an abelian subalgebra a of maximal dimension, and let I be a maximal subalgebra containing a that is an ideal of g . (i) Is it true that dim Z(I)≥n−2k+ 1 ? (ii) Is it true that dim C2(I)≤k−1 ? (iii) If they are true, do 1 and 2 imply that β(g) = n−k ? In reference to Chapter 3 , we would like to study the isomorphism classes of liform Lie algebras starting from some relation over the coecients αi and αj−1 j that were dened at the end of this chapter. Another open problem that we want to deal with is to compute the matrix representation of liform Lie algebras of dimension greater than or equal to 9 . We will also keep studying new methods to improve our representations and to obtain new ones for other types of Lie algebras such as solvable ones, for instance. 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(2) 49 :1, (1948) 142. 124 LIST OF FIGURES List of Tables 1 The invariant α for simple Lie algebras . . . . . . . . . . . . . . . . . vii 2.1 Lie algebras of dimension 4 whose α invariant is 2 ............ 19 3.1 Triples for non-model liform Lie algebras of dimension less than 12 . 54 3.2 Triples for non-model liform Lie algebras of dimension 12 and 13 . . . 55 3.3 Triples for non-model liform Lie algebras of dimension 14 ....... 56 3.4 Computing time and memory used. . . . . . . . . . . . . . . . . . . 56 4.1 Computing time and memory usage for allabsub ............ 71 4.2 Computing time and memory usage for allabideal . ......... 71 4.3 Complexity and number of operations. . . . . . . . . . . . . . . . . . 73 4.4 Lie algebras of dimension less than 5 . . . . . . . . . . . . . . . . . . 82 4.5 5 -dimensional non-decomposable real solvable non-nilpotent Lie algebras .................................... 83 4.6 5 -dimensional non-decomposable real solvable non-nilpotent Lie algebras(II).................................. 84 4.7 Cases when α and β invariants are dierent in non-decomposable real solvable Lie algebras of dimension 6 . . . . . . . . . . . . . . . . . . . 84 4.8 6 -dimensional non-decomposable solvable non-nilpotent Lie algebras . 85 4.9 6 -dimensional non-decomposable solvable non-nilpotent Lie algebras (II)..................................... 86 4.10 6 -dimensional non-decomposable solvable non-nilpotent Lie algebras (III) .................................... 87 4.11 6 -dimensional non-decomposable solvable non-nilpotent Lie algebras (IV) .................................... 88 4.12 6 -dimensional non-decomposable solvable non-nilpotent Lie algebras (V)..................................... 89 4.13 6 -dimensional non-decomposable solvable non-nilpotent Lie algebras (VI) .................................... 90 125 126 LIST OF TABLES 4.14 6 -dimensional non-decomposable solvable non-nilpotent Lie algebras (VII).................................... 91 4.15 α invariant for nilpotent Lie algebras of dimension less than 6 . . . . . 92 4.16 α invariant for 6 -dimensional nilpotent Lie algebras . . . . . . . . . . 92 4.17 6 -dimensional non-decomposable nilpotent Lie algebras . . . . . . . . 93 4.18 7 -dimensional non-decomposable nilpotent Lie algebras (I) . . . . . . 93 4.19 7 -dimensional non-decomposable nilpotent Lie algebras (II) . . . . . . 94 4.20 7 -dimensional non-decomposable nilpotent Lie algebras (III) . . . . . 95 4.21 7 -dimensional non-decomposable nilpotent Lie algebras (IV) . . . . . 96 4.22 Minimal faithful matrix representations for dimension ≤7 ......107 4.23 Minimal faithful matrix representations for dimension 8 (I) . . . . . . 108 4.24 Minimal faithful matrix representations for dimension 8 (II) . . . . . 109 List of authors Bianchi, v Bowman, ix, x Bratzlavsky, viii Cartan, vi, vii Cellini, vii, xi de Graaf, 74, 78 Elduque, ix, x Farnsteiner, ix Frajria, xi Garland, xii Gein, ix Ito, x Jacobson, vi, viii, 32, 36 Killing, vi, vii Kirillov, ix Konstant, vi Kostant, vii, xi, xii Kubo, viii, ix Kuznecov, ix Lepowsky, xii Levi, vi Magnin, 74, 78, 80 Malcev, vi, viii Milentyeva, xii, xiii Mubarakzyanov, 74, 75 Orsina, vii Panyushev, xi Papi, vii, xi Peterson, vii Petravchuk, x Petrov, v Röhrle, xi Romanovskii, xii Schur, vi, viii, 32, 36 Seeley, 74, 78 Shestakov, xii Stewart, viii Stitzinger, ix Suprunenko, viii Suter, xi Towers, ix, x Turkowski, 74, 75 Tyshkevich, viii Varea, ix, x Vergne, viii Zaicev, ix 127