On the degenerations of solvable Leibniz algebras

Abstract

The present paper is devoted to the description of rigid solvable Leibniz algebras. In particular, we prove that solvable Leibniz algebras under some conditions on the nilradical are rigid and we describe four-dimensional solvable Leibniz algebras with three-dimensional rigid nilradical. We show that the Grunewald-O'Halloran's conjecture "any n-dimensional nilpotent Lie algebra is a degeneration of some algebra of the same dimension" holds for Lie algebras of dimensions less than six and for Leibniz algebras of dimensions less than four. The algebra of level one, which is omitted in the 1991 Gorbatsevich's paper, is indicated.

The classification of algebras of level two

“…, x n } be a basis such that x 1 = x, x 2 ∈ S α and x 3 ∈ S −α with α = 0. Then [x 1 , x 2 ] = αx 2 and [x 1 , x 3 ] = −αx 3 . By scaling of basis elements we can assume that α = 1.…”

“…Degenerations of non-associative algebras were the subject of numerous papers (see for instance [2,3,6,9] and references given therein), and their research continues actively.…”

Classification of algebras of level two in the variety of nilpotent algebras and Leibniz algebras

“…Degenerations of Lie and Leibniz algebras were the subject of numerous papers, see for instance [1,4,5,10,11] and references given therein, and their research continues actively. In particular, in [6,15] some irreducible components of Leibniz algebras are found.…”

The Algebraic and Geometric Classification of Nilpotent Bicommutative Algebras

“…It is typical to focus on small dimensions, and there are two main directions for the classification: algebraic and geometric. Varieties as Jordan, Lie, Leibniz or Zinbiel algebras have been studied from these two approaches ( [1, 9, 12-15, 22, 27, 30, 37] and [3,5,6,9,11,22,24,25,31,32,[35][36][37][38], respectively). In the present paper, we give the algebraic and geometric classification of 4-dimensional nilpotent bicommutative algebras.…”