Central extensions of filiform Zinbiel algebras

Central extensions of filiform Zinbiel algebras 1 Luisa M. Camachoa, Iqboljon Karimjanovb,Ivan Kaygorodovc& Abror Khudoyberdiyevd aUniversity of Sevilla, Sevilla, Spain. bAndijan State University, Andijan, Uzbekistan. cCMCC, Universidade Federal do ABC, Santo Andr´e, Brazil. dNational University of Uzbekistan, Institute of Mathematics Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan. E-mail addresses: Luisa M. Camacho (lca[email protected]) Iqboljon Karimjanov (iqbol[email protected]om) Ivan Kaygorodov ([email protected]) Abror Khudoyberdiyev ([email protected]) Abstract:In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an n-dimensional nullfiliform Zinbiel algebra is isomorphic to an (n+1)-dimensional null-filiform Zinbiel algebra. Moreover, we obtain all pairwise non isomorphic quasi-filiform Zinbiel algebras. Keywords:Zinbiel algebra, filiform algebra, algebraic classification, central extension. MSC2010: 17D25, 17A30. INTRODUCTION The algebraic classification (up to isomorphism) of an n-dimensional algebras from a certain variety defined by some family of polynomial identities is a classical problem in the theory of non-associative algebras. There are many results related to algebraic classification of small dimensional algebras in the varieties of Jordan, Lie, Leibniz, Zinbiel and many another algebras [1,9,11–16,24,27,31,33,36–38,42]. An algebra Ais called a Zinbiel algebra if it satisfies the identity (x◦y)◦z=x◦(y◦z+z◦y). Zinbiel algebras were introduced by Loday in [43] and studied in [2,5,10,17,20–22,41,44–46,49]. Under the Koszul duality, the operad of Zinbiel algebras is dual to the operad of Leibniz algebras. Hence, the tensor product of a Leibniz algebra and a Zinbiel algebra can be given the structure of a Lie algebra. Under the symmetrized product, a Zinbiel algebra becomes an associative and commutative algebra. Zinbiel algebras are also related to Tortkara algebras [20] and Tortkara triple systems [7]. More precisely, 1This work was supported by Agencia Estatal de Investigaci´on (Spain), grant MTM2016-79661-P (European FEDER support included, UE); RFBR 20-01-00030; FAPESP 18/15712-0, 18/12197-7, 19/00192-3. 1 2 every Zinbiel algebra with the commutator multiplication gives a Tortkara algebra (also about Tortkara algebras, see, [24–26]), which have recently sprung up in unexpected areas of mathematics [18,19]. Central extensions play an important role in quantum mechanics: one of the earlier encounters is by means of Wigner’s theorem which states that a symmetry of a quantum mechanical system determines an (anti-)unitary transformation of a Hilbert space. Another area of physics where one encounters central extensions is the quantum theory of conserved currents of a Lagrangian. These currents span an algebra which is closely related to so-called affine Kac-Moody algebras, which are universal central extensions of loop algebras. Central extensions are needed in physics, because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra. Kac-Moody algebras have been conjectured to be symmetry groups of a unified superstring theory. The centrally extended Lie algebras play a dominant role in quantum field theory, particularly in conformal field theory, string theory and in M-theory. In the theory of Lie groups, Lie algebras and their representations, a Lie algebra extension is an enlargement of a given Lie algebra gby another Lie algebra h. Extensions arise in several ways. There is a trivial extension obtained by taking a direct sum of two Lie algebras. Other types are a split extension and a central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. A central extension and an extension by a derivation of a polynomial loop algebra over a finite-dimensional simple Lie algebra gives a Lie algebra which is isomorphic to a non-twisted affine Kac-Moody algebra [6, Chapter 19]. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra, the Heisenberg algebra is the central extension of a commutative Lie algebra [6, Chapter 18]. The algebraic study of central extensions of Lie and non-Lie algebras has a very long history [3, 28–30, 35, 39, 47, 48, 50]. For example, all central extensions of some filiform Leibniz algebras were classified in [3, 48] and all central extensions of filiform associative algebras were classified in [35]. Skjelbred and Sund used central extensions of Lie algebras for a classification of low dimensional nilpotent Lie algebras [47]. After that, the method introduced by Skjelbred and Sund was used to describe all non-Lie central extensions of all 4-dimensional Malcev algebras [30], all non-associative central extensions of 3-dimensional Jordan algebras [29], all anticommutative central extensions of 3-dimensional anticommutative algebras [8]. Note that the method of central extensions is an important tool in the classification of nilpotent algebras. It was used to describe all 4-dimensional nilpotent associative algebras [15], all 4-dimensional nilpotent assosymmetric algebras [32], all 4-dimensional nilpotent bicommutative algebras [40], all 4-dimensional nilpotent Novikov algebras [34], all 4-dimensional commutative algebras [23], all 5-dimensional nilpotent Jordan algebras [27], all 5-dimensional nilpotent restricted Lie algebras [14], all 5-dimensional anticommutative algebras [23], all 6-dimensional nilpotent Lie algebras [13,16], all 6-dimensional nilpotent Malcev algebras [31], all 6-dimensional nilpotent binary Lie algebras [1], all 6-dimensional nilpotent anticommutative CD-algebras [1], all 6-dimensional nilpotent Tortkara algebras [24,26], and some others. 3 1. PRELIMINARIES All algebras and vector spaces in this paper are over C. 1.1. Filiform Zinbiel algebras. An algebra Ais called Zinbiel algebra if for any x, y, z ∈Ait satisfies the identity (x◦y)◦z=x◦(y◦z) + x◦(z◦y). For an algebra A, we consider the series A1=A,Ai+1 = i X k=1 AkAi+1−k, i ≥1. We say that an algebra Ais nilpotent if Ai= 0 for some i∈N. The smallest integer satisfying Ai= 0 is called the nilpotency index of A. Definition 1. An n-dimensional algebra Ais called null-filiform if dim Ai= (n+1)−i, 1≤i≤n+1. It is easy to see that a Zinbiel algebra has a maximal nilpotency index if and only if it is null-filiform. For a nilpotent Zinbiel algebra, the condition of null-filiformity is equivalent to the condition that the algebra is one-generated. All null-filiform Zinbiel algebras were described in [4]. Throughout the paper, Cj idenotes the combinatorial numbers i j. Theorem 2. [4] An arbitrary n-dimensional null-filiform Zinbiel algebra is isomorphic to the algebra F0 n: ei◦ej=Cj i+j−1ei+j,2≤i+j≤n, where omitted products are equal to zero and {e1, e2,...,en}is a basis of the algebra. As an easy corollary from the previous theorem we have the next result. Theorem 3. Every non-split central extension of F0 nis isomorphic to F0 n+1. Proof. It is easy to see, that every non-split central extension of F0 nis a one-generated nilpotent algebra. It follows that every non-split central extension of a null-filiform Zinbiel algebra is a null-filiform Zinbiel algebra. Using the classification of null-filiform algebras (Theorem 2) we have the statement of the Theorem.  Definition 4. An n-dimensional algebra is called filiform if dim(Ai) = n−i, 2≤i≤n. All filiform Zinbiel algebras were classified in [4]. Theorem 5. An arbitary n-dimensional (n≥5) filiform Zinbiel algebra is isomorphic to one of the following pairwise non-isomorphic algebras: F1 n:ei◦ej=Cj i+j−1ei+j,2≤i+j≤n−1; F2 n:ei◦ej=Cj i+j−1ei+j,2≤i+j≤n−1, en◦e1=en−1; F3 n:ei◦ej=Cj i+j−1ei+j,2≤i+j≤n−1, en◦en=en−1. 4 1.2. Basic definitions and methods. Throughout this paper, we are using the notations and methods well written in [29, 30] and adapted for the Zinbiel case with some modifications. From now, we will give only some important definitions. Let (A,◦)be a Zinbiel algebra and Va vector space. Then the C-linear space Z2(A,V)is defined as the set of all bilinear maps θ:A×A−→ V, such that θ(x◦y, z) = θ(x, y ◦z+z◦y). Its elements will be called cocycles. For a linear map ffrom Ato V, if we write δf :A×A−→ Vby δf(x, y) = f(x◦y), then δf ∈Z2(A,V). We define B2(A,V) = {θ=δf :f∈Hom (A,V)}. One can easily check that B2(A,V)is a linear subspace of Z2(A,V)whose elements are called coboundaries. We define the second cohomology space H2(A,V)as the quotient space Z2(A,V)B2(A,V). Let Aut (A)be the automorphism group of the Zinbiel algebra Aand let φ∈Aut (A). For θ∈ Z2(A,V)define φθ(x, y) = θ(φ(x), φ (y)). Then φθ ∈Z2(A,V). So, Aut (A)acts on Z2(A,V). It is easy to verify that B2(A,V)is invariant under the action of Aut (A)and so we have that Aut (A)acts on H2(A,V). Let Abe a Zinbiel algebra of dimension m < n, and Vbe a C-vector space of dimension n−m. For any θ∈Z2(A,V)define on the linear space Aθ:= A⊕Vthe bilinear product “ [−,−]Aθ” by [x+x′, y +y′]Aθ=x◦y+θ(x, y)for all x, y ∈A, x′, y′∈V. The algebra Aθis a Zinbiel algebra which is called an (n−m)-dimensional central extension of Aby V. Indeed, we have, in a straightforward way, that Aθis a Zinbiel algebra if and only if θ∈Z2(A,V). We also call the set Ann (θ) = {x∈A:θ(x, A) + θ(A, x) = 0}the annihilator of θ. We recall that the annihilator of an algebra Ais defined as the ideal Ann (A) = {x∈A:x◦A+A◦x= 0}and observe that Ann (Aθ) = (Ann (θ)∩Ann (A)) ⊕V. We have the next key result: Lemma 6. Let Abe an n-dimensional Zinbiel algebra such that dim(Ann (A)) = m6= 0. Then there exists, up to isomorphism, a unique (n−m)-dimensional Zinbiel algebra A′and a bilinear map θ∈Z2(A,V)with Ann (A)∩Ann (θ) = 0, where Vis a vector space of dimension m, such that A∼ =A′θand A/Ann (A)∼ =A′. However, in order to solve the isomorphism problem we need to study the action of Aut (A)on H2(A,V). To do that, let us fix e1,...,esa basis of V, and θ∈Z2(A,V). Then θcan be uniquely written as θ(x, y) = s X i=1 θi(x, y)ei, where θi∈Z2(A,C). Moreover, Ann (θ) = Ann (θ1)∩Ann (θ2)∩ ...∩Ann (θs). Further, θ∈B2(A,V)if and only if all θi∈B2(A,C). Definition 7. Let Abe an algebra and Ibe a subspace of Ann(A). If A=A0⊕Ithen Iis called an annihilator component of A. 5 Definition 8. A central extension of an algebra Awithout annihilator component is called a non-split central extension. It is not difficult to prove (see [30, Lemma 13]), that given a Zinbiel algebra Aθ, if we write as above θ(x, y) = s X i=1 θi(x, y)ei∈Z2(A,V)and we have Ann (θ)∩Ann (A) = 0, then Aθhas an annihilator component if and only if [θ1],[θ2],...,[θs]are linearly dependent in H2(A,C). Let Vbe a finite-dimensional vector space. The Grassmannian Gk(V)is the set of all k-dimensional linear subspaces of V. Let Gs(H2(A,C)) be the Grassmannian of subspaces of dimension sin H2(A,C). There is a natural action of Aut (A)on Gs(H2(A,C)). Let φ∈Aut (A). For W=h[θ1],[θ2],...,[θs]i ∈ Gs(H2(A,C)) define φW =h[φθ1],[φθ2],...,[φθs]i. Then φW ∈ Gs(H2(A,C)). We denote the orbit of W∈Gs(H2(A,C)) under the action of Aut (A)by Orb(W). Since given W1=h[θ1],[θ2],...,[θs]i, W2=h[ϑ1],[ϑ2],...,[ϑs]i ∈ GsH2(A,C), we easily have that in case W1=W2, then s T i=1 Ann (θi)∩Ann (A) = s T i=1 Ann (ϑi)∩Ann (A), and so we can introduce the set Ts(A) = (W=h[θ1],[θ2],...,[θs]i ∈ GsH2(A,C): s \ i=1 Ann (θi)∩Ann (A) = 0), which is stable under the action of Aut (A). Now, let Vbe an s-dimensional linear space and let us denote by E(A,V)the set of all non-split s-dimensional central extensions of Aby V. We can write E(A,V) = (Aθ:θ(x, y) = s X i=1 θi(x, y)eiand h[θ1],[θ2],...,[θs]i ∈ Ts(A)). We also have the next result, which can be proved as in [30, Lemma 17]. Lemma 9. Let Aθ,Aϑ∈E(A,V). Suppose that θ(x, y) = s X i=1 θi(x, y)eiand ϑ(x, y) = s X i=1 ϑi(x, y)ei. Then the Zinbiel algebras Aθand Aϑare isomorphic if and only if Orb h[θ1],[θ2],...,[θs]i= Orb h[ϑ1],[ϑ2],...,[ϑs]i. From here, there exists a one-to-one correspondence between the set of Aut (A)-orbits on Ts(A)and the set of isomorphism classes of E(A,V). Consequently we have a procedure that allows us, given the Zinbiel algebra A′of dimension n−s, to construct all non-split central extensions of A′. This procedure would be: 6 Procedure (1) For a given Zinbiel algebra A′of dimension n−s, determine H2(A′,C),Ann (A′)and Aut (A′). (2) Determine the set of Aut (A′)-orbits on Ts(A′). (3) For each orbit, construct the Zinbiel algebra corresponding to a representative of it. Finally, let us introduce some of notation. Let Abe a Zinbiel algebra with a basis e1, e2,...,en. Then by ∆i,j we will denote the bilinear form ∆i,j :A×A−→ Cwith ∆i,j (el, em) = δilδjm. Then the set {∆i,j : 1 ≤i, j ≤n}is a basis for the linear space of the bilinear forms on A. Then every θ∈Z2(A,C) can be uniquely written as θ=X 1≤i,j≤n cij∆i,j, where cij ∈C. 2. CENTRAL EXTENSION OF FILIFORM ZINBIEL ALGEBRAS Proposition 10. Let F1 n, F2 nand F3 nbe n-dimensional filiform Zinbiel algebras defined in Theorem 5. Then: •A basis of Z2(Fk n,C)is formed by the following cocycles Z2(F1 n,C) = h∆1,1,∆1,n,∆n,1,∆n,n, s−1 P i=1 Ci−1 s−1∆i,s−i; 3 ≤s≤ni, Z2(Fk n,C) = h∆1,1,∆1,n,∆n,1,∆n,n, s−1 P i=1 Ci−1 s−1∆i,s−i; 3 ≤s≤n−1i, k = 2,3. •A basis of B2(Fk n,C)is formed by the following coboundaries B2(F1 n,C) = h∆1,1, s−1 P i=1 Ci−1 s−1∆i,s−i,3≤s≤n−1i, B2(F2 n,C) = h∆1,1, s−1 P i=1 Ci−1 s−1∆i,s−i,3≤s≤n−2, n−2 P i=1 Ci−1 n−2∆i,n−1−i+ ∆n,1i, B2(F3 n,C) = h∆1,1, s−1 P i=1 Ci−1 s−1∆i,s−i,3≤s≤n−2, n−2 P i=1 Ci−1 n−2∆i,n−1−i+ ∆n,ni. •A basis of H2(Fk n,C)is formed by the following cocycles H2(F1 n,C) = h[∆1,n],[∆n,1],[∆n,n],[ n−1 P i=1 Ci−1 n−1∆i,n−i]i, H2(Fk n,C) = h[∆1,n],[∆n,1],[∆n,n]i, k = 2,3. Proof. The proof follows directly from the definition of a cocycle.  Proposition 11. Let φn k∈Aut (Fk n). Then 7 φn 1=         a1,10 0 . . . 0 0 a2,1a2 1,10. . . 0 0 a3,1∗a3 1,1. . . 0 0 . . .. . .. . ..... . .. . . an−1,1∗ ∗ an−1 1,1an−1,n an,10 0 . . . 0an,n         , φn 2=         a1,10 0 ... 0 0 a2,1a2 1,10... 0 0 a3,1∗a3 1,1... 0 0 . . .. . .. . ..... . .. . . an−1,1∗ ∗ an−1 1,1an−1,n an,10 0 ... 0an−2 1,1         , φn 3=          a1,10 0 ... 0 0 a2,1a2 1,10... 0 0 a3,1∗a3 1,1... 0 0 . . .. . .. . ..... . .. . . an−1,1∗ ∗ an−1 1,1an−1,n an,10 0 ... 0a(n−1)/2 1,1          2.1. Central extensions of F1 n.Let us denote ∇1= [∆1,n],∇2= [∆n,1],∇3= [∆n,n],∇4= [ n−1 X j=1 Cj−1 n−1∆j,n−j] and x=a1,1, y =an,n, z =an−1,n, w =an,1. Since         ∗... ∗C0 n−1α′ 4α′ 1 ∗. . . C1 n−1α′ 40 0 . . .... . . . . . . . . . Cn−2 n−1α′ 4... 0 0 0 α′ 2... 0 0 α′ 3         = (φn 1)T                    0 0 0 ··· 0C0 n−1α4α1 0 0 0 ··· C1 n−1α40 0 0 0 0 . . .·· · 0 0 . . . . . ....Cn−1−i n−1α4·· ·. . . . . . . . .0·· · 0 . . . . . . . . . 0Cn−3 n−1α40 . . . . . . . . . . . . Cn−2 n−1α40 0 ··· 0 0 0 α20 0 ··· 0 0 α3                    φn 1, for any θ=α1∇1+α2∇2+α3∇3+α4∇4, we have the action of the automorphism group on the subspace hθias D(α1xy +α3yw +α4xz)∇1+ (α2xy +α3yw + (n−1)α4xz)∇2+α3y2∇3+α4xn∇4E. 2.1.1. 1-dimensional central extensions of F1 n.Let us consider the following cases: (1) if α16= 0, α2=α3=α4= 0,then by choosing x= 1, y = 1/α1,we have the representative h∇1i. (2) if α26= 0, α3=α4= 0,then by choosing x= 1, y = 1/α2, α =α1/α2,we have the family of representatives hα∇1+∇2i. (3) if α1=α2, α36= 0, α4= 0,then by choosing y= 1/√α3, w =−α2/α3, x = 1,we have the representative h∇3i. 8 (4) if α16=α2, α36= 0, α4= 0,then by choosing x=√α3 α1−α2, y =1 √α3, w =α2 √α3(α2−α1),we have the representative h∇1+∇3i. (5) if (n−1)α1=α2, α3= 0, α46= 0,then by choosing x= 1/n √α4, y = 1, z =−α1/α4,we have the representative h∇4i. (6) if (n−1)α16=α2, α3= 0, α46= 0,then by choosing x= 1/n √α4, y =n √α4 α2−(n−1)α1, z = −n √α4 α2−(n−1)α1,we have the representative h∇2+∇4i. (7) if α36= 0, α46= 0,then by choosing x= 1/n √α4, y = 1/√α3, z =α1−α2 (n−2)√α3α4, w =α2−(n−1)α1 (n−2) n √α4α3, we have the representative h∇3+∇4i. It is easy to verify that all previous orbits are different, and so we obtain T1(F1 n) = Orbh∇1i∪Orbhα∇1+∇2i∪Orbh∇3i∪Orbh∇1+∇3i∪ Orbh∇4i∪Orbh∇2+∇4i∪Orbh∇3+∇4i. 2.1.2. 2-dimensional central extensions of F1 n.We may assume that a 2-dimensional subspace is generated by θ1=α1∇1+α2∇2+α3∇3+α4∇4, θ2=β1∇1+β2∇2+β3∇3. Then we have the six following cases: (1) if α46= 0, β36= 0,then we can suppose that α3= 0.Now (a) for (n−1)α16=α2, β16=β2,by choosing x=(α2−(n−1)α1)(β2−β1) α41/(n−2), y =β2−β1 β3x, z=α1(β1−β2) α4β3x, w =−β1x/β3,we have the representative h∇2+∇3,∇2+∇4i. (b) for (n−1)α16=α2, β1=β2,by choosing x=α2−(n−1)α1 α4√β31/(n−1), y = 1/√β3, z = −α1 α4√β3, w =−β1x/β3,we have the representative h∇3,∇2+∇4i. (c) for (n−1)α1=α2, β16=β2,by choosing x= 1/n √α4, y =β2−β1 β3x, z =α1(β1−β2) α4β3x, w=−β1x/β3,we have the representative h∇2+∇3,∇4i. (d) for (n−1)α1=α2, β1=β2,by choosing x= 1/n √α4, y = 1/√β3, z =−α1y/α4, w=−β1x/β3,we have the representative h∇3,∇4i. (2) if α46= 0, β3= 0, β26= 0,then we can suppose that α2= 0.Now (a) for α36= 0,by choosing x= 1/n √α4, y = 1/√α3, z =α1 (n−2)√α3α4, w =−(n−1)α1 (n−2) n √α4α3,and α=β1/β2we have the family of representatives hα∇1+∇2,∇3+∇4i. (b) for α3= 0,(n−1)β16=β2,then by choosing x= 1/n √α4, y = 1, z =−α1β2 α4((n−1)β1−β2)and α=β1/β2,we have the family of representatives hα∇1+∇2,∇4iα6=1 n−1. (c) for α3= 0,(n−1)β1=β2and α1= 0,by choosing x= 1/n √α4, y = 1, z = 0,we have the representative h1 n−1∇1+∇2,∇4i. 9 (d) for α3= 0,(n−1)β1=β2and α16= 0,by choosing x= 1/n √α4, y =−n √α4 (n−1)α1, z = n √α4 (n−1)α4,we have the representative h1 n−1∇1+∇2,∇2+∇4i. (3) if α46= 0, β3=β2= 0, β16= 0,then (a) for α36= 0,by choosing x= 1/n √α4, y = 1/√α3, z =α1−α2 (n−2)√α3α4, w =α2−(n−1)α1 (n−2) n √α4α3,we have the representative h∇1,∇3+∇4i. (b) for α3= 0,after a linear combination of θ1and θ2we can suppose that (n−1)α1=α2,by choosing x= 1/n √α4, y = 1, z =−α1/α4,we have the representative h∇1,∇4i. (4) if α4= 0, α36= 0, β26= 0,then (a) for β16=β2,after a linear combination of θ1and θ2we can suppose that α1=α2,by choosing y= 1/√α3, w =−α2/α3, x = 1 and α=β1/β2we have the family of representatives hα∇1+∇2,∇3iα6=1. (b) for β1=β2, α1=α2,after a linear combination of θ1and θ2we have the representative h∇1+∇2,∇3i. (c) for β1=β2, α16=α2,by choosing x=√α3 α1−α2, y = 1/√α3, w =α2 √α3(α2−α1),we have the representative h∇1+∇2,∇1+∇3i. (5) if α4= 0, α36= 0, β2= 0, β16= 0,then after a linear combination of θ1and θ2we can suppose that α1=α2,by choosing y= 1/√α3, w =−α2/α3, x = 1 and α=β1/β2we have the representative h∇1,∇3i. (6) if α3=α4= 0, β3= 0, then we have the representative h∇1,∇2i. It is easy to verify that all previous orbits are different, and so we obtain T2(F1 n) = Orbh∇1,∇2i∪Orbh∇1,∇3i∪Orbh∇1,∇3+∇4i∪Orbh∇1,∇4i∪ Orbh1 n−1∇1+∇2,∇2+∇4i∪Orbh∇1+∇2,∇1+∇3i∪Orbhα∇1+∇2,∇3i∪ Orbhα∇1+∇2,∇3+∇4i∪Orbhα∇1+∇2,∇4i∪Orbh∇2+∇3,∇2+∇4i∪ Orbh∇2+∇3,∇4i∪Orbh∇3,∇2+∇4i∪Orbh∇3,∇4i. 2.1.3. 3-dimensional central extensions of F1 n.We may assume that a 3-dimensional subspace is generated by θ1=α1∇1+α2∇2+α3∇3+α4∇4, θ2=β1∇1+β2∇2+β3∇3, θ3=γ1∇1+γ2∇2. 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