Central extensions of null-filiform and naturally graded filiform non-lie Leibniz algebras

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1 Journal of Algebra 479 (207) Contents lists available at ScienceDirect Journal of Algebra Central extensions of null-filiform and naturally graded filiform non-lie Leibniz algebras J.K. Adashev a, L.M. Camacho b,, B.A. Omirov a a Institute of Mathematics, National University of Uzbekistan, Tashkent, 0025, Uzbekistan b Dpto. Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes, s/n, 402 Sevilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 23 February 206 Available online 6 February 207 Communicated by Vera Serganova MSC: 7A32 7B30 Keywords: Leibniz algebra Filiform algebra Quasi-filiform algebra Natural gradation Characteristic sequence 2-cocycles Central extension In this paper we describe central extensions of some nilpotent Leibniz algebras. Namely, central extensions of the Leibniz algebra with maximal index of nilpotency are classified. Moreover, non-split central extensions of naturally graded filiform non-lie Leibniz algebras are described up to isomorphism. It is shown that k-dimensional central extensions (k 5) of these algebras are split. 207 Elsevier Inc. All rights reserved. * Corresponding author. addresses: adashevjq@mail.ru (J.K. Adashev), lcamacho@us.es (L.M. Camacho), omirovb@mail.ru (B.A. Omirov) / 207 Elsevier Inc. All rights reserved.

2 462 J.K. Adashev et al. / Journal of Algebra 479 (207) Introduction 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 the universal central extension 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 a 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 g by 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 split extension and 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 finite-dimensional simple Lie algebra give a Lie algebra which is isomorphic with a non-twisted affine Kac Moody algebra [2, Chapter 9]. 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 [2, Chapter 8], [3,2]. During last 20 years Leibniz algebras are studied mainly to extend the classical results from Lie algebras. From structural point of view some classical analogues which are true for nilpotent Lie algebras are obtained. In particular, papers devoted to the study of naturally graded nilpotent Leibniz algebras are [,4 7] and many others. In the present paper we consider a notion of a central extension of Leibniz algebras, which is similar to the notion of a central extension of Lie algebras []. Here we consider central extensions of some nilpotent Leibniz algebras, such as null-filiform Leibniz algebra and naturally graded non-lie filiform Leibniz algebras. In fact, the method of central extensions of Lie algebras is adapted for Leibniz algebras in [9]. In works [9,0] one-dimensional central extensions of the null-filiform Leibniz algebra and the naturally graded filiform Lie algebra are described. The difficulties of the description of one-dimensional central extensions of a naturally graded Lie algebra show that the further progress in description of filiform Lie algebras by using central extensions is too complicated. That is why in our study we restrict to non-lie Leibniz algebras.

3 J.K. Adashev et al. / Journal of Algebra 479 (207) The main results of this paper consist of the classification of central extensions of null-filiform Leibniz algebras and naturally graded filiform non-lie Leibniz algebras. Throughout the paper all vector spaces and algebras are finite-dimensional over the field of complex numbers. Moreover, in the table of multiplication of an algebra the omitted products are assumed to be zero. 2. Preliminaries Let F be an algebraically closed field of characteristic 0. Definition 2.. [8] An algebra L over a field F is called a Leibniz algebra if for any elements x, y, z L the so-called Leibniz identity is satisfied: where [, ] is a multiplication in L. [x, [y, z]] = [[x, y],z] [[x, z],y], Let L be a finite-dimensional Leibniz algebra. For this algebra we define sequence: L = L, L k+ =[L k,l ], k. A Leibniz algebra L is called nilpotent if there exists s N such that L s = {0}. The minimal number s possessing of such property is called index of nilpotency of the algebra L. Definition 2.2. [] A Leibniz algebra L of dimension n is called null-filiform if diml i = (n +) i, i n +. Evidently, by definition algebra being null-filiform is equivalent that it has maximal possible index of nilpotency. Theorem 2.3. [] In arbitrary n-dimensional null-filiform Leibniz algebra there exists a basis {e, e 2,..., e n } such that the table of multiplication of the algebra is of the following form: NF n : [e i,e ]=e i+, i n. Definition 2.4. [] A Leibniz algebra L is called filiform if diml i = n i, 2 i n. Note that the nilindex of a filiform Leibniz algebras coincides with its dimension. Let L be a Leibniz algebra of nilindex s. We put L i = L i /L i+, i s. Then we obtain a graded algebra GrL = L L 2 L s, where [L i, L j ] L i+j. An algebra L is called a naturally graded algebra if L = GrL. In the following theorem we summarize the results of the works [,3].

4 464 J.K. Adashev et al. / Journal of Algebra 479 (207) Theorem 2.5. An arbitrary complex n-dimensional naturally graded filiform Leibniz algebra is isomorphic to one of the following pairwise non-isomorphic algebras: F 3 n : F n : [e,e ]=e 3, [e i,e ]=e i+, 2 i n, Fn 2 : [e,e ]=e 3, [e i,e ]=e i+, 3 i n, { [ei,e ]= [e,e i ]=e i+, 2 i n [e i,e n+ i ]= [e n+ i,e i ]=α( ) i+ e n, 2 i n, where α {0, } for even n and α =0for odd n. It is known that the set of filiform Leibniz algebras is decomposed into three disjoint classes, two families of non-lie Leibniz algebras which arise from the algebras F n and F 2 n and one family of algebras (which include filiform Lie algebras) arises from the Lie algebra F 3 n [7]. Since further we consider only filiform non-lie Leibniz algebras we omit the third family of algebras. Theorem 2.6. Any (n +)-dimensional complex non-lie filiform Leibniz algebra, whose naturally graded algebra is not a Lie algebra, belongs to one of the following two classes: [e,e ]=e 3, [e i,e ]=e i+, 2 i n, Fn(α 4,α 5,...,α n,θ): [e,e 2 ]= n α k e k + θe n, k=4 [e i,e 2 ]= n k=i+2 [e,e ]=e 3, α k+ i e k, 2 i n 2, [e i,e ]=e i+, 3 i n, Fn(β 2 4,β 5,...,β n,γ): [e,e 2 ]= n β k e k, k=4 [e 2,e 2 ]=γe n, [e i,e 2 ]= n β k+ i e k, 3 i n 2. k=i+2 Definition 2.7. An algebra L is called quasi-filiform if L n 2 {0} and L n = {0}, where diml = n. Let x be a nilpotent element of the set L \ L 2. For the nilpotent operator of right multiplication R x we define a decreasing sequence C(x) = (n, n 2,..., n k ), which consists of the dimensions of Jordan blocks of the operator R x. In the set of such sequences we con-

5 J.K. Adashev et al. / Journal of Algebra 479 (207) sider the lexicographic order, that is, C(x) = (n, n 2,..., n k ) C(y) = (m, m 2,..., m s ) if and only if there exists i N such that n j = m j for any j <iand n i <m i. Definition 2.8. The sequence C(L) = max C(x) x L\L 2 of the algebra L. is called characteristic sequence Definition 2.9. A quasi-filiform non-lie Leibniz algebra L is called algebra ( of type I, ) if there exists an element x L\L 2 Jn 2 0 such that the operator R x has the form:. 0 J 2 Combining the results of the papers [4] and [6] we have the following theorem. Theorem 2.0. Let L be a naturally graded quasi-filiform Leibniz algebra of type I. Then it is isomorphic to the one of the following pairwise non-isomorphic algebras: NF n 2 C 2 ; Fn C; [e i,e ]=e i+, i n 3, { L n : [e,e n ]=e n + e 2, L 2 [e i,e ]=e i+, i n 3, n : [e,e n ]=e n. [e i,e n ]=e i+, 2 i n 3. [e i,e ]=e i+, i n 3 [e i,e ]=e i+, i n 3 L,λ n : [e n,e ]=e n, L 2,λ [e n,e ]=e n, n : [e,e n ]=λe n, λ {0, } [e,e n ]=λe n, λ C [e n,e n ]=e n [e i,e ]=e i+, i n 3 [e i,e ]=e i+, i n 3 L 3,λ n : [e n,e ]=e n + e 2, L 4,λ n : [e n,e ]=e n + e 2, [e,e n ]=λe n, λ {, 0, } [e n,e n ]=λe n, λ 0 [e i,e ]=e i+, i n 3 L 5,λ,μ [e n,e ]=e n + e 2, n : [e,e n ]=λe n, (λ, μ) =(, ) or (2, 4) [e n,e n ]=μe n, [e i,e ]=e i+, i n 3 [e n,e ]=e n, L 6 n : [e,e n ]= e n, [e n,e n ]=e 2, [e n,e n ]=e 3. Central extensions of Leibniz algebras. It is remarkable that any nilpotent Leibniz algebra has non-trivial center. Let L be a Leibniz algebra over a field F and Z(L) be its center.

6 466 J.K. Adashev et al. / Journal of Algebra 479 (207) The central extension of a Leibniz algebra L by V is called a short exact sequence of Leibniz algebras 0 V ε L λ L 0, where Imε = Kerλ and V is the center of the algebra L. Two central extensions 0 V ɛ L λ L 0 and 0 V ɛ L λ L 0 are called equivalent if there exists an isomorphism ϕ : L L such that ϕ ɛ = ɛ, λ ϕ = λ. A bilinear map θ : L L V satisfying an identity θ(x, [y, z]) = θ([x, y],z) θ([x, z],y), for any elements x, y, z L is called central 2-cocycle. We denote by ZL 2 (L, V )the set of all central 2-cocycles from L to V. If for a linear map ϕ : L V we have θ(x, y) = ϕ([x, y]), then θ is called 2-coboundary. We denote by BL 2 (L, V )the set of all 2-coboundaries from L to V. The quotient space HL 2 (L, V ) := ZL 2 (L, V )/BL 2 (L, V )is called the 2-nd group of cohomology. For a given 2-cocycle θ on L we construct the associated central extension L θ. By definition it is a vector space L θ = L V with equipped multiplication: [x + u, y + v] =[x, y] L + θ(x, y), () where [, ] L is the bracket on L and x, y L, u, v V. If θ θ 2 is a 2-coboundary, then θ and θ 2 define the same central extension. In this case, θ and θ 2 are called cohomological. 3. Classification of k-dimensional central extensions of null-filiform Leibniz algebras In this section, we focus on k-dimensional central extensions of null-filiform Leibniz algebra. First, we compute a basis of HL 2 (NF n, V ). Proposition 3.. Let L be the null-filiform Leibniz algebra and let V = <x,x 2,...,x k > be an abelian algebra. Then: A basis of ZL 2 (NF n, V ) is formed by the following cocycles θ i,j (e i,e )=x j, i n, j k

7 J.K. Adashev et al. / Journal of Algebra 479 (207) A basis of BL 2 (NF n, V ) is formed by the following coboundaries θ i,j (e i,e )=x j, i n, j k A basis of HL 2 (NF n, V ) is formed by the following cocycles θ j (e n,e )=x j, j k Proof. The proof follows directly from the definition of a cocycle. Theorem 3.2. A k-dimensional central extension of an n-dimensional complex nullfiliform Leibniz algebra is isomorphic to one of the following non-isomorphic algebras: NF n C k, NF n+ C k. Proof. Let {e, e 3,..., e n } be a basis of NF n and {x,..., x k } be a basis of V. From the definition of the product on central extension () we conclude that {e, e 3,..., e n, x, x 2,..., x k } forms a basis of the algebra L = NF n V. Applying Proposition 3., we get that the table of multiplication of the central extension of NF n has the following form: [e i,e ]=e i+, i n, [e n,e ]= k α i x i. i= If α i =0for all i with i k, then we obtain the algebra NF n C k. If there exists α i 0for some i, then setting e n+ = NF n+ C k. k i= α i x i we get the algebra 4. Classification of k-dimensional central extensions of naturally graded non-lie filiform Leibniz algebras In this section we present the classification of k-dimensional central extensions of naturally graded non-lie filiform Leibniz algebras. Let V = <x,x 2,...,x k > be an abelian algebra. First, we compute a basis of HL 2 (F i n, V )with i =, 2. Proposition 4.. The following cocycles θ i,j (e i,e )=x j, i n, θ n+,j (e,e 2 )=x j, θ n+2,j (e 2,e 2 )=x j form a basis of ZL 2 (F n, V ) and ZL 2 (F 2 n, V );

8 468 J.K. Adashev et al. / Journal of Algebra 479 (207) The following coboundaries a) θ,j (e, e ) = x j, θ,j (e 2, e ) = x j, θ i,j (e i+, e ) = x j, 2 i n 2, j k, form a basis of BL 2 (F n, V ), b) θ,j (e, e ) = x j, θ i,j (e i+, e ) = x j, 2 i n 2, j k, form a basis of BL 2 (F 2 n, V ); Basis of the spaces HL 2 (F n, V ) and HL 2 (F 2 n, V ) are formed by the following cocycles θ,j (e 2,e )=x j, θ 2,j (e n,e )=x j, θ 3,j (e,e 2 )=x j, θ 4,j (e 2,e 2 )=x j, j k. Proof. The proof is carried out by straightforward calculations. One-dimensional central extensions. In this subsection we study the classification of one-dimensional central extensions of the algebras F n and F 2 n. Theorem 4.2. An arbitrary one-dimensional central extension of the complex algebra F n is isomorphic to one of the following pairwise non-isomorphic algebras: F n C; F n+(0,...,0,α 4,α 3 ), α 3,α 4 {0, }; L 3,λ n+ ; L4,λ n+ ; L5,λ,μ n+ ; L n+. Proof. From Proposition 4. and the construction of central extensions, we derive that the table of multiplication of one-dimensional extension of F n has the form: [e,e ]=e 3, [e i,e ]=e i+, 3 i n, [e 2,e ]=e 3 + α x, [e n,e ]=α 2 x, [e,e 2 ]=α 3 x, [e 2,e 2 ]=α 4 x. We distinguish the following cases: If α i =0for any i with i 4, then we get the algebra F n C. If there exists i such that α i 0, then we consider the cases. Case. Let α 2 0, then one can assume α 2 =. Note that diml i = n + i for 2 i n +. Consequently we have an (n + )-dimensional filiform Leibniz algebra. Making the following change of basis e 2 = e 2 α e n and e n+ = x, we derive α =0. Hence, in this case we obtain the family of algebras F n+(0, 0,..., 0, α 4, α 3 ). In the work [7], the authors show that instead of the general change of basis in the family F n+(α 4, α 5,..., α n+, θ) it is sufficient to consider the following change of basis: e = Ae + Be 2, e 2 =(A + B)e 2 + B(α 3 α 4 )e n, e 3 = A(A + B)e 3 +(ABα 3 + B 2 α 4 )e n+, e i = A i 2 (A + B)e i, 4 i n +, A+ B 0.

9 J.K. Adashev et al. / Journal of Algebra 479 (207) From [e, e 2] = α 3e n+ and [e 2, e 2] = α 4e n+, we obtain Consider the following subcases. α 3 = Aα 3 + Bα 4 A n, α 4 = (A + B)α 4 A n. a) α 4 =0. Then α 4 =0and α 3 = α 3 A n 2. If α 3 = 0, then α 3 = 0. Otherwise, putting A = n 2 α 3 we have that α 3 =. Therefore, we have F n+(0,..., 0, 0, α 3 ), where α 3 {0, }. b) α 4 0. Putting B = An Aα 4 α 4, we obtain α 4 =and α 3 = α 3 α 4 +A n 2 A n 2. If α 3 = α 4, then α 3 =, otherwise, putting A = n 2 α 4 α 3, we get α 3 = 0. Therefore, we obtain the algebras F n+(0,..., 0,, α 3 ), where α 3 {0, }. Thus, we have F n+(0,..., 0, α 4, α 3 ), where α 3, α 4 {0, }. Case 2. Let α 2 =0. Taking the change of basis we get the algebra e = e,e i = e i+, 2 i n, e n = e 2,e n+ = x, [e i,e ]=e i+, i n 2, [e n,e ]=e 2 + α e n+, [e,e n ]=α 3 e n+, [e n,e n ]=α 4 e n+. Note that this algebra is an (n + )-dimensional quasi-filiform Leibniz algebra. Case 2.. α 0. If α 4 =0, then we get the algebra which is isomorphic to L 3,λ n+. If α 3 =0, α 4 0, then the obtained algebra is isomorphic to L 4,λ n+. If α 3 0, α 4 0, then the derived algebra is isomorphic to L 5,λ,μ n+. Case 2.2. α =0. Take the change of basis e = e + e n,e 2 =2e 2 +(α 3 + α 4 )e n+,e i =2e i, 3 i n, e n+ = e n+. Then we obtain the algebra [e i,e ]=e i+, i n 2, [e n,e ]=e 2 +(α 4 α 3 )e n+, [e,e n ]=2(α 3 + α 4 )e n+, [e n,e n ]=4α 4 e n+. We can consider α 3 = α 4 0 because in other case we are α 0. Therefore, we may assume that α 3 = α 4 0. Making the change of basis e i = e i, i n, e n = e e n,e n+ = 4α 3 e n+, we deduce the algebra L n+.

10 470 J.K. Adashev et al. / Journal of Algebra 479 (207) Note that in the case the obtained algebras are filiform and in the case 2 are quasi-filiform. The next theorem describes one-dimensional central extensions of the algebra F 2 n. Theorem 4.3. An arbitrary one-dimensional central extension of the algebra F 2 n is isomorphic to one of the following pairwise non-isomorphic algebras: F 2 n C; F 2 n+; F 2 n+(0,...,0, 0, ); F 2 n+(0,...,0,, 0); L,λ n+ ; L2,λ n+ ; L2 n+. Proof. From Proposition 4. and from the construction of the central extensions, we derive that the table of multiplication of one-dimensional extension of F 2 n has the form: [e,e ]=e 3, [e i,e ]=e i+, 3 i n, [e 2,e ]=α x, [e n,e ]=α 2 x, [e,e 2 ]=α 3 x, [e 2,e 2 ]=α 4 x. If α i =0for any i with i 4, then we get a split algebra F 2 n C. If α i 0for some i, then we consider the possible cases. Case. Let α 2 0be. We can assume α 2 =. Since diml i = n + i for 2 i n +, we have an (n + )-dimensional filiform Leibniz algebra. Making the change e 2 = e 2 α e n and e n+ = x, we obtain α =0. So, in this case we get the family F 2 n+(0, 0,..., 0, α 3, α 4 ). Arguing similarly as in the proof of Theorem 4.2, we obtain the following pairwise non-isomorphic algebras: F 2 n+, F 2 n+(0,...,0,, 0), F 2 n+(0,...,0, 0, ). Case 2. Let α 2 =0. Making the following change of basis: we get the family of algebras e = e, e i = e i+, 2 i n, e n = e 2, e n+ = x, [e i,e ]=e i+, i n 2, [e n,e ]=α e n+, [e,e n ]=α 3 e n+, [e n,e n ]=α 4 e n+. Note that these algebras are (n + )-dimensional quasi-filiform Leibniz algebras of type I. Case 2.. Let α 0. We can assume α =. If α 4 =0, then the obtained algebra is isomorphic to the algebra L,λ If α 4 0, then the derived algebra is isomorphic to the algebra L 2,λ n+. n+.

11 J.K. Adashev et al. / Journal of Algebra 479 (207) Case 2.2. Let α =0. Take the change of basis e = e + e n,e 2 = e 2 +(α 3 + α 4 )e n+,e i = e i, 3 i n +. Then we obtain the following family of algebras: [e i,e ]=e i+, i n 2, [e n,e ]=α 4 e n+, [e,e n ]=(α 3 + α 4 )e n+, [e n,e n ]=α 4 e n+. It is easy to see that if α 4 0, then we are in conditions of the case α 0. Therefore, we assume that α 4 =0and α 3 0. Then making the change of basis e n+ = α 3 e n+, we get L 2 n+. Two-dimensional central extensions. This subsection is devoted to the study of classification of two-dimensional non-split central extensions of the algebras F n and F 2 n. First, we consider two-dimensional central extensions of the algebra F n. From Proposition 4. and from the definition of central extensions, we conclude that the table of multiplication of two-dimensional central extension of F n has the form: [e,e ]=e 3, [e 2,e ]=e 3 + α x + β x 2, [e i,e ]=e i+, 3 i n, [e n,e ]=α 2 x + β 2 x 2, [e,e 2 ]=α 3 x + β 3 x 2, [e 2,e 2 ]=α 4 x + β 4 x 2. Let us take the following change of basis e = e,e 2 = e 3,e i = e i+, 2 i n, e n = e 2, e n+ = x,e n+2 = x 2. Then we obtain: [e i,e ]=e i+, i n 2, [e n,e ]=e 2 + α e n+ + β e n+2, [e n,e ]=α 2 e n+ + β 2 e n+2, [e,e n ]=α 3 e n+ + β 3 e n+2, [e n,e n ]=α 4 e n+ + β 4 e n+2. Theorem 4.4. An arbitrary two-dimensional non-split central extension of the algebra F n is isomorphic to one of the following pairwise non-isomorphic algebras: L(, 0, 0, 0), L(,, 0, 0), L(0,, 0, 0), L(0,,, 0), L(, 0, 0, β 4 ), β 4 0, L(0,,, 0), L(0,,, ), L(0,, 2, 4), where

12 472 J.K. Adashev et al. / Journal of Algebra 479 (207) [e i,e ]=e i+, i n, [e n+,e ]=e 2 + e n+2, L(α 3,α 4,β 3,β 4 ) [e,e n+ ]=α 3 e n + β 3 e n+2, [e n+,e n+ ]=α 4 e n + β 4 e n+2, M(, β 4 ), β 4, M(0, 0), M(0, ), M(0, 2), M(2, ), M(2, 2), M(, ), where [e i,e ]=e i+, i n 2, [e n,e ]=e 2 + e n+, M(α 4,β 4 ): [e,e n ]=e n+2, [e n,e n ]=α 4 e n+ + β 4 e n+2, L n+2, L 3,λ n+2, L4,λ n+2, L5,λ,μ n+2 and the algebra Proof. We distinguish the following cases: [e i,e ]=e i+, i n, L [e,e n+ ]=e 2 + e n+2, : [e i,e n+ ]=e i+, 2 i n, [e n+,e n+ ]=e n. Case. Let (α 2, β 2 ) (0, 0). Without loss of generality, one can assume α 2 0and [e n, e ] = e n+. Making the change e n = e n α e n, we obtain α =0. Setting e n = e n+, e n+ = e n, we have: [e i,e ]=e i+, i n, [e n+,e ]=e 2 + β e n+2, [e,e n+ ]=α 3 e n + β 3 e n+2, [e n+,e n+ ]=α 4 e n + β 4 e n+2. Case.. Let β i =0for i 4. Then we get a split central extension. Case.2. Let β 0. Putting e n+2 = β e n+2, we can assume that β =. Since diml 2 = n and diml i = n + i for 3 i n, we have an (n + 2)-dimensional quasi-filiform Leibniz algebra with characteristic sequence equal to C(L) = (n, 2). Let us take the general change of basis: n+2 n+2 e = A i e i, e n+ = B i e i. i= Then we express the new basic elements {e, e 2,..., e n+2} via the basic elements {e, e 2,..., e n+2 }. Then for e 2 we have i=

13 J.K. Adashev et al. / Journal of Algebra 479 (207) n 2 e 2 =[e,e ]=A (A + A n+ )e 2 + A A i e i+ + i=2 +(A A n + A A n+ α 3 + A 2 n+α 4 )e n +(A A n+ + A A n+ β 3 + A 2 n+β 4 )e n+2. Further, we obtain the expressions for e t with 3 t n, namely: n t+ e t = A t (A + A n+ )e t + A t A i e i+t, 3 t n, i=2 e n = A n (A + A n+ )e n. Therefore, A (A + A n+ ) 0. From the equations [e n, e n+] = A n 2 (A + A n+ )B e n =0we derive B =0. Since e n+2 =[e n+, e ] e 2, we deduce n 2 e n+2 =(A B n+ A (A + A n+ ))e 2 + A (B i A i )e i+ + +[(A B n + A n+ B n+ α 4 ) (A A n + A A n+ α 3 + A 2 n+α 4 )]e n + +[(A B n+ + A n+ B n+ β 4 ) (A A n+ + A A n+ β 3 + A 2 n+β 4 )]e n+2. Take into account that in new basis [e n+2, e ] = 0, we derive the restrictions: Considering the products i=2 B n+ = A + A n+, B i = A i, 2 i n 2. [e,e n+] =α 3e n + β 3e n+2, [e n+,e n+] =α 4e n + β 4e n+2, we derive the expressions for the new parameters: α 3 = A2 α 3 + A A n+ α 4 + A n+ B n+ (α 3 β 4 α 4 β 3 )+A (A n B n )β 3 + A n+ (A n B n )β 4 A n, (A + A n+ β 4 A n+ β 3 ) () α 4 = B n+(a α 4 + A n+ (α 3 β 4 α 4 β 3 )+(A n B n )β 4 ) A n, (2) (A + A n+ β 4 A n+ β 3 ) β 3 = (A β 3 + A n+ β 4 )B n+ A (A + A n+ β 4 A n+ β 3 ), (3) β 4 = β 4 B 2 n+ A (A + A n+ β 4 A n+ β 3 ), (4) with condition A (A + A n+ )(A + A n+ β 4 A n+ β 3 ) 0.

14 474 J.K. Adashev et al. / Journal of Algebra 479 (207) Consider now the possible cases. Case.2.. Let β 4 =0. Then β 4 =0and we obtain that β 3 is nullity invariant, that is, β 3 =0if and only if β 3 =0. Thus, we can distinguish two cases. a) Let β 3 =0. a.) α 4 =0. Then α 3 = α 3 A n 2, α 4 =0. If α 3 =0, then α 3 =0. Otherwise, putting A = n 2 α 3. Hence, we obtain α 3 =. In this subcase we get the algebras L 3,0 n+2 and L(, 0, 0, 0). a.2) α 4 0. Then α 3 = A α 3 +A n+ α 4, α A n 4 = (A +A n+ )α 4. Note that the nullity of A n the following expression α 3 α 4 = α 3 α 4 A n 2 If α 3 α 4 =0, then putting A n+ = An is invariant. A α 4 α 4 algebra L(,, 0, 0). If α 3 α 4 0, then choosing A n+ = A α 3 α 4 α 3 =0, α 4 =. Therefore, we get the algebra L(0,, 0, 0). b) Let β 3 0. Then putting we obtain, we have α 3 = α 4 =and the and A = n 2 α 4 α 3, we obtain A n B n = A2 α 3 + A A n+ α 4 A n+ B n+ α 4 β 3 A β 3 α 3 =0, α 4 = (A + A n+ )α 4 A n, β 3 = (A + A n+ )β 3. A A n+ β 3 From the last expression we deduce that β 3 += A (β 3 +) A A n+ β 3 is nullity invariant. We distinguish the following subcases: b.) β 3. Putting A n+ = A ( β 3 ) 2β 3, we obtain β 3 =, α 4 = (β 3+)α 4 If α 4 =0, then α 4 =0and we have the algebra L 3, n+2.. 2A n 2 β 3 If α 4 0, then choosing A = n 2 (β3 +)α 4 2β 3, we obtain α 4 =and the algebra L(0,,, 0). b.2) β 3 =. It implies β 3 =, α 4 = (A +A n+ )α 4. A n If α 4 =0, then α 4 =0and the algebra L 3, n+2 is obtained. If α 4 0, then setting A n+ = An A α 4 α 4, we get α 4 =and the algebra L(0,,, 0). Case.2.2. Let β 4 0. We have either (β 3, β 4 ) = (0, β 4 )with β 4 0or (β 3, β 4 ) = (, ) or (β 3, β 4 ) = (2, 4), as it had in the case of naturally graded quasi-filiform Leibniz algebras (see the proof of Theorem 9 in [6]).

15 J.K. Adashev et al. / Journal of Algebra 479 (207) a) Let (β 3, β 4 ) = (0, β 4 )with β 4 0. Then from the expressions () (4) we get α 3 = α 3 A n, α 4 = A α 4 +(A n B n )β 4 A n, A n+ =0. Taking A n B n = A α 4 β 4, we obtain α 4 =0. If α 3 =0, then we have α 3 =0and we obtain the algebra L 4,λ n+2 with λ 0. If α 3 0, choosing A = n α 3, we have α 3 =and the algebra L(, 0, 0, β 4 ), β 4 0. b) Let (β 3, β 4 ) = (, ). Then expressions () (4) imply α 3 = A α 3 + A n B n A n, α 4 = A α 4 +(A n B n ) A n, A n+ =0. Setting A n B n = A α 3, we deduce α 3 =0and α 4 = α 4 α 3. A n 2 Note that the following expression α 3 α 4 = α 3 α 4 A n 2 is nullity invariant. If α 3 α 4 =0, then we obtain α 4 =0and the algebra L 5,, n+2. If α 3 α 4 0, then choosing A = n 2 α 4 α 3, we have α 4 =and the algebra L(0,,, ). c) Let (β 3, β 4 ) = (2, 4). Then we have: α 3 = A α 3 +2(A n B n ) A n, α 4 = A α 4 +4(A n B n ) A n, A n+ =0. Putting A n B n = A α 3 2, we derive α 3 =0and α 4 = α 4 2α 3. A n 2 It is easy to check that the following expression: 2α 3 α 4 = 2α 3 α 4 A n 2 is nullity invariant. If α 3 2α 4 =0, then we have α 4 =0and the algebra L 5,2,4 n+2. If α 3 2α 4 0, then choosing A = n 2 α 4 2α 3, we obtain α 4 =and the algebra L(0,, 2, 4). Case.3. Let β =0. Let us make the following change of basis: e = e +(α 4 α 3 )e n + e n+,e 2 =2e 2 +2α 4 e n +(β 3 + β 4 )e n+2, e i =2e i, 3 i n +,e n+2 = e n+2.

16 476 J.K. Adashev et al. / Journal of Algebra 479 (207) Then the table of multiplication of the algebra has the form: [e i,e ]=e i+, i n, [e n+,e ]=e 2 +(β 4 β 3 )e n+2, [e,e n+ ]=(α 3 + α 4 )e n +2(β 3 + β 4 )e n+2, [e n+,e n+ ]=2α 4 e n +4β 4 e n+2. We have β 3 = β 4 0. Indeed, if β 3 β 4, then we are in the case of β 0. Putting e n+2 =(α 3 +α 4 )e n +4β 3 e n+2 we obtain an (n +2)-dimensional quasi-filiform Leibniz algebra with C(L) = (n,, ): [e i,e ]=e i+, i n, [e n+,e ]=e 2, [e,e n+ ]=e n+2, [e n+,e n+ ]=α 4 e n + e n+2. Taking the general change of generator basis elements and applying similar arguments as in Case.2 we derive the expression for the new parameter: α 4 = α 4 A n 2 Since α 4 is nullity invariant, we can easily get that α 4 is equal either to 0 or to. Therefore, we have the algebras: [e i,e ]=e i+, i n, [e n+,e ]=e 2, [e n+,e n+ ]=e n+2, [e,e n+ ]=e n+2, The following change of basis:. [e i,e ]=e i+, i n, [e n+,e ]=e 2, [e n+,e n+ ]=e n + e n+2, [e,e n+ ]=e n+2. e i = e i, i n, e n+ = e e n+,e n+2 = e n+2 leads to the algebras L n+2 and L. Case 2. Let (α 2, β 2 ) = (0, 0) and (α, β ) (0, 0). Without loss of generality, we might assume that α 0. Therefore, taking e n+ = e 2 + α e n+ + β e n+2, we get [e n, e ] = e 2 + e n+. Thus, we have the table of multiplications: [e i,e ]=e i+, i n 2, [e n,e ]=e 2 + e n+, [e,e n ]=α 3 e n+ + β 3 e n+2, [e n,e n ]=α 4 e n+ + β 4 e n+2.

17 J.K. Adashev et al. / Journal of Algebra 479 (207) Note that diml 2 = n, diml i = n + i, 3 i n. Case 2.. Let β 3 0. Putting e n+2 = α 3 e n+ +β 3 e n+2, we can assume α 3 =0, β 3 =. Then we have the family of algebras: [e i,e ]=e i+, i n 2, [e n,e ]=e 2 + e n+, M(α 4,β 4 ): [e,e n ]=e n+2, [e n,e n ]=α 4 e n+ + β 4 e n+2. Similar to Case.2 we get the expressions for the new parameters with the restriction Consider the following: α 4 = β 4 = B 2 nα 4 A 2 + A A n β 4 + A A n α 4 + A 2 nα 4, B n (A β 4 + A n α 4 ) A 2 + A A n β 4 + A A n α 4 + A 2 nα 4, A (A + A n )(A 2 + A A n β 4 + A A n α 4 + A 2 nα 4 ) 0. Let α 4 =0. Then α 4 =0and β 4 = (A +A n )β 4 A +A n β 4. If β 4 =0, then we obtain the algebra M(0, 0). If β 4 0, then we have β 4 = (A +A n )β 4 A +A n β 4. Note that the expression β 4 = A (β 4 ) A +A n β 4 implies nullity invariant. Therefore the assumption β 4 = deduces the algebra M(0, ) and in opposite assumption (that is, β 4 ), choosing A n = A (β 4 2) β 4, we obtain β 4 =2and the algebra M(0, 2). Let α 4 0. The following expressions: α 4 β 4 = A (A + A n )(α 4 β 4 ) A 2 + A A n β 4 + A A n α 4 + A 2 nα 4, β 4 A 2 = (β 4 ) A 2 + A A n β 4 + A A n α 4 + A 2 nα 4 give us nullity invariants. a) α 4 β 4 0and β 4. Putting A n = A ( α 4 ) α 4 β 4, we obtain the algebra M(, β 4 ), where β 4. b) α 4 β 4 0and β 4 =. Setting A n = (α 4 2)A α 4, we have α 4 =2and the algebra M(2, ). c) α 4 β 4 =0and β 4. Choosing A n = A ( α 4 + 2α 2 4 2α 4) α 4, we get α 4 =2and the algebra M(2, 2). d) α 4 β 4 =0and β 4 =. Then we obtain α 4 = β 4 =and the algebra M(, ).

18 478 J.K. Adashev et al. / Journal of Algebra 479 (207) Case 2.2. Let β 3 =0and β 4 0. Let us take the change of basis as follows: e = A e + A n e n, e n =(A + A n )e n, with A (A + A n ) 0. Similar to Case.2 we get e n+ =[e n, e ] e 2 =(A 2 + A A n α 4 A A n α 3 )e n+ + A A n β 4 e n+2. On the other hand, we have [e, e n] =(A + A n )(A α 3 + A n α 4 )e n+ + (A + A n )A n β 4 e n+2. Consider the determinant: A 2 + A A n α 4 A A n α 3 A A n β 4 (A + A n )(A α 3 + A n α 4 ) (A + A n )A n β 4 = A A n (A + A n )β 4 (A A n α 3 A α 3 ) 0. Therefore, we conclude that e n+ and [e, e n]are lineally independent. Therefore, we are in the conditions of Case 2.. Case 2.3. Let (α, β ) = (0, 0). Taking the following change of basis: e = e + e n,e 2 =2e 2 +(α 3 + α 4 )e n+ +(β 3 + β 4 )e n+2, we obtain the family of algebras: e i =2e i, 3 i n, e n+ = e n+,e n+2 = e n+2, [e i,e ]=e i+, i n 2, [e n,e ]=e 2 +(α 4 α 3 )e n+ +(β 4 β 3 )e n+2, [e,e n ]=2(α 3 + α 4 )e n+ +2(β 3 + β 4 )e n+2, [e n,e n ]=4α 4 e n+ +4β 4 e n+2. Since (α, β ) = (0, 0), we can assume that (α 3, β 3 ) = (α 4, β 4 ). Making the change e n+ =4α 3 e n+ +4β 3 e n+2, we obtain a split algebra. Now, we study the two-dimensional non-split central extensions of the algebra F 2 n. From Proposition 4. and definition of central extension, we conclude that the table of multiplication of two-dimensional central extension of the algebra F 2 n has the following form: [e,e ]=e 3, [e 2,e ]=α x + β x 2, [e i,e ]=e i+, 3 i n, [e n,e ]=α 2 x + β 2 x 2, [e,e 2 ]=α 3 x + β 3 x 2, [e 2,e 2 ]=α 4 x + β 4 x 2. It we take the change of basis e = e,e 2 = e 3,e i = e i+, 2 i n, e n = e 2,e n+ = x,e n+2 = x 2,

19 J.K. Adashev et al. / Journal of Algebra 479 (207) then its multiplication has the following form: [e i,e ]=e i+, i n 2, [e n,e ]=α e n+ + β e n+2, [e n,e ]=α 2 e n+ + β 2 e n+2, [e,e n ]=α 3 e n+ + β 3 e n+2, [e n,e n ]=α 4 e n+ + β 4 e n+2. Theorem 4.5. An arbitrary two-dimensional non-split central extension of the algebra F 2 n is isomorphic to one of the following pairwise non-isomorphic algebras: N(, 0, 0, 0), N(0, α 4, β 3, 0), α 4 {0, }, β 3 0, N(, 0, 0, ), N(0,, 0, 0), N(0,,, ), where [e i,e ]=e i+, i n, [e n+,e ]=e n+2, N(α 3,α 4,β 3,β 4 ): [e,e n+ ]=α 3 e n + β 3 e n+2, [e n+,e n+ ]=α 4 e n + β 4 e n+2, R(0, 0,, 0), R(0, 0,, ), R(0,,, β 4 ), β 4 C, R(, 0, 0, ) where [e i,e ]=e i+, i n 2, [e n,e ]=e n+, R(α 3,α 4,β 3,β 4 ): [e,e n ]=α 3 e n+ + β 3 e n+2, [e n,e n ]=α 4 e n+ + β 4 e n+2, L,λ n+2, L2,λ n+2, L2 n+2 and the algebra [e i,e ]=e i+, i n, N : [e,e n+ ]=e n+2, [e n+,e n+ ]=e n. Proof. The proof is similar to the proof of Theorem 4.4. Three-dimensional central extensions. In this subsection we study three-dimensional non-split central extensions of the algebras F n and F 2 n. From Proposition 4. and definition of central extension, we conclude that the table of multiplication of three-dimensional central extension of the algebra F n has the form:

20 480 J.K. Adashev et al. / Journal of Algebra 479 (207) [e,e ]=e 3, [e 2,e ]=e 3 + α x + β x 2 + γ x 3, [e i,e ]=e i+, 3 i n, [e n,e ]=α 2 x + β 2 x 2 + γ 2 x 3, [e,e 2 ]=α 3 x + β 3 x 2 + γ 3 x 3, [e 2,e 2 ]=α 4 x + β 4 x 2 + γ 4 x 3. Making the change of basis e = e,e i = e i+, 2 i n, e n = e 2,e n+j = x j, j 3, we obtain [e i,e ]=e i+, i n 2, [e n,e ]=e 2 + α e n+ + β e n+2 + γ e n+3, [e n,e ]=α 2 e n+ + β 2 e n+2 + γ 2 e n+3, [e,e n ]=α 3 e n+ + β 3 e n+2 + γ 3 e n+3, [e n,e n ]=α 4 e n+ + β 4 e n+2 + γ 4 e n+3. Theorem 4.6. An arbitrary three-dimensional non-split central extension of the algebra F n is isomorphic to one of the following pairwise non-isomorphic algebras: P (0, 0, 0), P (, 0, 0), P (0, 0, ), P (, 0, ), P (0, 0, 2), P (, 0, 2), P (0,, γ 4 ), γ 4, P (0, 2, ), P (0, 2, 2), P (0,, ), where [e i,e ]=e i+, i n, [e n+,e ]=e 2 + e n+2, P (α 4,β 4,γ 4 ): [e,e n+ ]=e n+3, [e n+,e n+ ]=α 4 e n + β 4 e n+2 + γ 4 e n+3, [e i,e ]=e i+, i n 2, P [e n,e ]=e 2 + e n+, : [e,e n ]=e n+2, [e n,e n ]=e n+3. Proof. Consider the following cases: Case. (α 2, β 2, γ 2 ) (0, 0, 0). Without loss of generality, one can assume that α 2 0 and [e n, e ] = e n+. Taking e n = e n α e n, we get α =0.

21 J.K. Adashev et al. / Journal of Algebra 479 (207) Making the following change of basis: e i = e i, i n, e n = e n+,e n+ = e n, e n+2 = e n+2, e n+3 = e n+3, we transform the table of multiplication to the form: [e i,e ]=e i+, i n, [e n+,e ]=e 2 + β e n+2 + γ e n+3, [e,e n+ ]=α 3 e n + β 3 e n+2 + γ 3 e n+3, [e n+,e n+ ]=α 4 e n + β 4 e n+2 + γ 4 e n+3. If β i =0or γ i =0for any i, then we obtain a split Leibniz algebra. Therefore, we consider the case of β i 0and γ j 0for some i and j. Case.. (β, γ ) (0, 0). Without loss of generality, one can assume that β 0and [e n+, e ] = e 2 + e n+2. Hence, we have the following algebra: with (γ 3, γ 4 ) (0, 0). [e i,e ]=e i+, i n, [e n+,e ]=e 2 + e n+2, [e,e n+ ]=α 3 e n + β 3 e n+2 + γ 3 e n+3, [e n+,e n+ ]=α 4 e n + β 4 e n+2 + γ 4 e n+3, γ 3 0. Then we may assume [e, e n+ ] = e n+3 (by replacing e n+3 = α 3 e n +β 3 e n+2 + γ 3 e n+3 ). Thus, we have the family of algebras: [e i,e ]=e i+, i n, [e n+,e ]=e 2 + e n+2, [e,e n+ ]=e n+3, [e n+,e n+ ]=α 4 e n + β 4 e n+2 + γ 4 e n+3. Taking the general base change and applying similar arguments as in Case.2 of Theorem 4.4 we obtain the expressions for the new parameters: α 4 = (A + A n+ )(A α 4 +(A n B n )β 4 ) A n 2 (A 2 + A A n+ β 4 + A A n+ γ 4 + A 2 n+ β 4), β 4 (A + A n+ ) 2 β 4 = A 2 + A A n+ β 4 + A A n+ γ 4 + A 2 n+ β, 4 γ 4 = (A + A n+ )(A γ 4 + A n+ β 4 ) A 2 + A A n+ β 4 + A A n+ γ 4 + A 2 n+ β, 4

22 482 J.K. Adashev et al. / Journal of Algebra 479 (207) with restriction A (A + A n+ )(A 2 + A A n+ β 4 + A A n+ γ 4 + A 2 n+β 4 ) 0. Consider all possible cases. a) β 4 =0. Then β 4 =0. a.) γ 4 =0. Then α 4 = (A +A n+ )α 4. A n If α 4 =0, then α 4 =0, otherwise, putting A n+ = An A α 4 α 4, we get α 4 =, and the algebras P (0, 0, 0), P (, 0, 0). a.2) γ 4 0. It is remarkable that the following expression γ 4 = A (γ 4 ) A +A n+ γ 4 deduces nullity invariant. If γ 4 =, then α 4 = α 4 A n 2, γ 4 =. If α 4 =0, then we have the algebra P (0, 0, ) and when α 4 0, putting A = n 2 α 4, we get the algebra P (, 0, ). If γ 4, then choosing A n+ = A (γ 4 2) γ 4, we deduce γ 4 =2and α 4 = 2α 4 γ 4 A n 2. In the case of α 4 =0, we have the algebra P (0, 0, 2). In the opposite case (that is α 4 0) choosing A = n 2 2α4 γ 4, we have α 4 =and the algebra P (, 0, 2). b) β 4 0. Putting A n B n = A α 4 β 4 we deduce α 4 =0. The equalities β 4 γ 4 A (A + A n+ )(β 4 γ 4 ) = A 2 + A A n+ β 4 + A A n+ γ 4 + A 2 n+ β, 4 γ 4 A 2 = (γ 4 ) A 2 + A A n+ β 4 + A A n+ γ 4 + A 2 n+ β, 4 add two more nullity invariants. Consider the following subcases: b.) β 4 γ 4 0and γ 4. Choosing A n+ = A ( β 4 ) β 4 γ 4, we get β 4 =, γ 4 = β 4 γ 4 γ 2 4 +β 4 β 2 4 β 4 ( γ 4 ) and the algebra P (0,, γ 4 ), where γ 4. b.2) β 4 γ 4 0and γ 4 =. Setting A n+ = (β 4 2)A β 4 we derive α 4 =2, γ 4 = and the algebra P (0, 2, ). b.3) β 4 γ 4 =0and γ 4. Putting A n+ = A ( β 4 + 2β4 2 2β 4) β 4, we obtain β 4 =2 and the algebra P (0, 2, 2). b.4) β 4 γ 4 =0and γ 4 =. Then we get β 4 = γ 4 =and the algebra P (0,, ). γ 3 =0. It implies γ 4 0. Thus, we have the following family of algebras: [e i,e ]=e i+, i n, [e n+,e ]=e 2 + e n+2, [e,e n+ ]=α 3 e n + β 3 e n+2, [e n+,e n+ ]=α 4 e n + β 4 e n+2 + γ 4 e n+3.

23 J.K. Adashev et al. / Journal of Algebra 479 (207) Let us take the general change of generator basis elements: e = A e + A n+ e n+, e n+ =(A + A n+ )e n+, with A (A + A n+ ) 0. Similar to above we derive: e 2 =[e,e ]=A (A + A n+ )e 2 +(A A n+ α 3 + A 2 n+α 4 )e n + +(A A n+ + A A n+ β 3 + A 2 n+β 4 )e n+2 + A 2 n+γ 4 e n+3 and Since e n+2 =[e n+, e ] e 2, then e t = A t (A + A n+ )e t, 3 t n. e n+2 =(A A n+ α 4 A A n+ α 3 )e n +(A 2 + A A n+ β 4 A A n+ β 3 )e n A A n+ γ 4 e n+3. We have [e,e n+] =(A + A n+ )(A α 3 + A n+ α 4 )e n + +(A + A n+ )(A β 3 + A n+ β 4 )e n+2 +(A + A n+ )A n+ γ 4 e n+3. Consider A 2 + A A n+ β 4 A A n+ β 3 A A n+ γ 4 (A + A n+ )(A β 3 + A n+ β 4 ) (A + A n+ )A n+ γ 4 = A A n+ (A + A n+ )γ 4 (A A n+ β 3 A β 3 ) 0. Therefore, we conclude that e n+2 and [e, e n+] are lineally independent, which satisfy the conditions of the case γ 3 0. Case.2. (β, γ ) = (0, 0). Note that β 3 γ 4 β 4 γ 3 0, because otherwise the algebra is split. Let us make the following change of basis: e = e +(α 4 α 3 )e n + e n+,e 2 =2e 2 +2α 4 e n +(β 3 + β 4 )e n+2 +(γ 3 + γ 4 )e n+3, e i =2e i, 3 i n +,e n+2 = e n+2,e n+3 = e n+3.

24 484 J.K. Adashev et al. / Journal of Algebra 479 (207) Then we obtain the following family of algebras: [e i,e ]=e i+, i n, [e n+,e ]=e 2 +(β 4 β 3 )e n+2 +(γ 4 γ 3 )e n+3, [e,e n+ ]=(α 3 + α 4 )e n +2(β 3 + β 4 )e n+2 +2(γ 3 + γ 4 )e n+3, [e n+,e n+ ]=2α 4 e n +4β 4 e n+2 +4γ 4 e n+3. The assumption (β, γ ) = (0, 0) implies β 3 = β 4 0and γ 3 = γ 4 0. But this is a contradiction with the restriction β 3 γ 4 β 4 γ 3 0. α β γ Case 2. (α 2, β 2, γ 2 ) = (0, 0, 0) and det α 3 β 3 γ 3 0. Without loss of generality α 4 β 4 γ 4 we can assume that α β 3 γ 4 0, from which we have the algebra P. Now, we present three-dimensional central extensions of the algebra F 2 n. Theorem 4.7. Three-dimensional non-split central extension of the algebra F 2 n is isomorphic to one of the following pairwise non-isomorphic algebras: Q(0, 0, 0,, 0), Q(, 0, 0,, 0), Q(0, 0, 0,, ), Q(, 0, 0,, ), Q(0, 0,,, γ 4 ), γ 4 C; Q(0,, 0, 0, ), where [e i,e ]=e i+, i n, [e n+,e ]=e n+2, Q(α 4,β 3,β 4,γ 3,γ 4 ): [e,e n+ ]=β 3 e n+2 + γ 3 e n+3, [e n+,e n+ ]=α 4 e n + β 4 e n+2 + γ 4 e n+3, [e i,e ]=e i+, i n 2, Q [e n,e ]=e n+, : [e,e n ]=e n+2, [e n,e n ]=e n+3. Proof. The proof is carried out similar to the proof of Theorem 4.6. Four-dimensional central extensions. This subsection is devoted to the classification of four-dimensional non-split central extensions of the algebras F n and F 2 n. Analogously, the table of multiplication of three-dimensional central extension of the algebra F n has the form:

25 J.K. Adashev et al. / Journal of Algebra 479 (207) [e,e ]=e 3, [e i,e ]=e i+, 3 i n, [e 2,e ]=e 3 + α x + β x 2 + γ x 3 + δ x 4, [e n,e ]=α 2 x + β 2 x 2 + γ 2 x 3 + δ 2 x 4, [e,e 2 ]=α 3 x + β 3 x 2 + γ 3 x 3 + δ 3 x 4, [e 2,e 2 ]=α 4 x + β 4 x 2 + γ 4 x 3 + δ 4 x 4. Making the change of basis we deduce e = e,e i = e i+, 2 i n, e n = e 2,e n+ = x,e n+2 = x 2,e n+3 = x 3,e n+4 = x 4, [e i,e ]=e i+, i n 2, [e n,e ]=e 2 + α e n+ + β e n+2 + γ e n+3 + δ e n+4, [e n,e ]=α 2 e n+ + β 2 e n+2 + γ 2 e n+3 + δ 2 e n+4, [e,e n ]=α 3 e n+ + β 3 e n+2 + γ 3 e n+3 + δ 3 e n+4, [e n,e n ]=α 4 e n+ + β 4 e n+2 + γ 4 e n+3 + δ 4 e n+4. Theorem 4.8. Four-dimensional non-split central extension of the algebra Fn is isomorphic to the following algebra: [e i,e ]=e i+, i n 2, [e n,e ]=e n+2, [e n,e ]=e 2 + e n+, [e,e n ]=e n+3, [e n,e n ]=e n+4. α β γ δ α 2 β 2 γ 2 δ 2 Proof. Note that det 0. Otherwise we get a split Leibniz algebra. α 3 β 3 γ 3 δ 3 α 4 β 4 γ 4 δ 4 Without loss of generality, one can assume α β 2 γ 3 δ 4 0. Thus, we obtain the algebra of the theorem. Similar to the case of F n, we obtain the following result: Theorem 4.9. Four-dimensional non-split central extension of the algebra F 2 n is isomorphic to the following algebra:

26 486 J.K. Adashev et al. / Journal of Algebra 479 (207) [e i,e ]=e i+, i n 2, [e n,e ]=e n+2, [e n,e ]=e n+, [e,e n ]=e n+3, [e n,e n ]=e n+4. Finally, we present the description of k-dimensional (k 5) central extensions of naturally graded filiform non-lie Leibniz algebras. In fact, all of them are split. Theorem 4.0. A k-dimensional (k 5) central extension of the algebras F n and F 2 n is a split algebra. Acknowledgments The authors were supported by Ministerio de Economía y Competitividad (Spain), grant MTM P and by V Plan Propio de Investigación of Sevilla University (VPPI-US). The last named author was also partially supported by the Grant No. 0828/GF4 of Ministry of Education and Science of the Republic of Kazakhstan. References [] Sh.A. Ayupov, B.A. Omirov, On some classes of nilpotent Leibniz algebras, Sibirsk. Mat. Zh. 42 () (200) 8 29 (in Russian); translation in Sib. Math. J. 42 () (200) [2] G.G.A. Bäuerle, E.A. de Kerf, A.P.E. ten Kroode, Lie Algebras. Part 2. Finite and Infinite Dimensional Lie Algebras and Applications in Physics, edited and with a preface by E.M. de Jager, Studies in Mathematical Physics, vol. 7, North-Holland Publishing Co., Amsterdam, ISBN , 997, x+554 pp. [3] M. Cahen, C. Ohn, Bialgebra structures on the Heisenberg algebra, Bull. Cl. Sci. Acad. Roy. Belg. 75 (989) [4] L.M. Camacho, J.R. Gómez, A.J. González, B.A. Omirov, Naturally graded 2-filiform Leibniz algebras, Comm. Algebra 38 (0) (200) [5] L.M. Camacho, J.R. Gómez, A.J. González, B.A. Omirov, The classification of naturally graded p-filiform Leibniz algebras, Comm. Algebra 39 () (20) [6] L.M. Camacho, J.R. Gómez, A.J. González, B.A. Omirov, Naturally graded quasi-filiform Leibniz algebras, J. Symbolic Comput. 44 (5) (2009) [7] J.R. Gómez, B.A. Omirov, On classification of complex filiform Leibniz algebras, Algebra Colloq. 22 () (205) [8] J.L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. 39 (993) [9] I.S. Rakhimov, S.L. Langari, M.B. Langari, On central extensions of null-filiform Leibniz algebras, Int. J. Algebra 3 (6) (2009) [0] I.S. Rakhimov, M.A. Hassan, On one-dimensional Leibniz central extensions of a filiform Lie algebra, Bull. Aust. Math. Soc. 84 (20) [] T. Skjelbred, T. Sund, On the Classification of Nilpotent Lie Algebras, Technical Report, Mathematisk Institutt, Universitetet i Oslo, 977. [2] I. Szymczak, S. Zakrzewski, Quantum deformation of the Heisenberg group obtained by geometric quantization, J. Geom. Phys. 7 (990) [3] M. Vergne, Cohomologie des algèbres de Lie nilpotentes. Application à l étude de la varieté des algèbres de Lie nilpotentes, Bull. Soc. Math. France 98 (970) 8 6.