A.Kh. Khudoyberdiyev

Leibniz algebras associated with representations of filiform Lie algebras

Abstract In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra n n , 1 . We introduce a Fock module for the algebra n n , 1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra n n , 1 with condition that the ideal I is a Fock n n , 1 -module, where I is the ideal generated by squares of elements from L . We also consider Leibniz algebras with corresponding Lie algebra n n , 1 and such that the action I × n n , 1 → I gives rise to a minimal faithful representation of n n , 1 . The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4 .

Infinitesimal deformations of null-filiform Leibniz superalgebras

Abstract In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras N F n , m . One of them is a Leibniz algebra (that is m = 0 ) and the second one is a pure Leibniz superalgebra (that is m ≠ 0 ) of maximum nilindex. We show that the closure of the union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable deformation of the algebra N F n . Similar results for the case of Leibniz superalgebras are obtained.

Infinitesimal deformations of naturally graded filiform Leibniz algebras

Abstract In the present paper we describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any n -dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra F n 3 ( 0 ) . We establish that in the same way any n -dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras F n 1 , F n 2 and F n 3 ( α ) . Moreover, we describe the linear integrable deformations of the above-mentioned algebras with a fixed basis of H L 2 in the set of all n -dimensional Leibniz algebras. Among these deformations one new rigid algebra has been found.

The classification of algebras of level two

Abstract This paper is devoted to the description of complex finite-dimensional algebras of level two. We obtain the classification of algebras of level two in the varieties of Jordan, Lie and associative algebras.

Complex Nilpotent Leibniz Superalgebras with Nilindex Equal to Dimension

We present the description up to isomorphism of Leibniz superalgebras with the characteristic sequence equal to (n | m 1,…, m k ) and nilindex n + m, where m = m 1 + … +m k , and n and m (m ≠ 0) are the dimensions of the even and odd parts, respectively.