1.1 Graded associative algebras. A nite Z-grading of an algebra A is a decomposition A = L n i=?n A i such that A i A j A i+j , where A i = 0 for jij > n. From now on a grading means a nite Z-grading. I am interested in associative graded algebras because they arise naturally in study of Lie algebras, although the subject is certainly interesting in its own right. A classiication of gradings of...

In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from the Zassenhaus algebra tensored with the divided powers algebra. We completely determine such class of algebras, calculating in process low-dimensional cohomol...

The L∞-algebra controlling simultaneous deformations of a morphism of associative algebras and its domain and codomain is described. Isomorphism of the cohomology of this L∞-algebra with the classical construction is shown.

A basis for a relatively free associative algebra with the identity x3 = 0 over a field of an arbitrary characteristic is found. As an application, a minimal generating system for the 3× 3 matrix invariant algebra is determined.

In this paper, by using Gröbner–Shirshov bases for non-associative algebras invented by A. I. Shirshov in 1962, we show I. P. Shestakov’s result that any Akivis algebra can be embedded into its universal enveloping algebra.

The purpose of this paper is to define cohomology complexes and study deformation theory Hom-associative algebra morphisms Hom-Lie morphisms. We discuss infinitesimal deformations, e...

By using the approach to Hall algebras arising in homologically finite triangulated categories in [9], we find an “almost” associative multiplication structure for indecomposable objects in a 2-period triangulated category. As an application, we give a new proof of the theorem of Peng and Xiao in [6] which provides a way of constructing all symmetrizable Kac-Moody Lie algebras from two periodic...

In order to describe non-Hamiltonian (dissipative) systems in quantum theory we need to use non-Lie algebra such that commutators of this algebra generate Lie subalgebra. It was shown that classical connection between analytic group (Lie group) and Lie algebra, proved by Lie theorems, exists between analytic loop, commutant of which is associative subloop (group), and commutant Lie algebra (an ...