Degenerations of Zinbiel and nilpotent Leibniz algebras

Abstract

), Alexandre Pozhidaev (app@math.nsc.ru), Yury Volkov (wolf86 666@list.ru).Abstract. We describe degenerations of four-dimensional Zinbiel and four-dimensional nilpotent Leibniz algebras over C. In particular, we describe all irreducible components in the corresponding varieties.

The algebraic classification of nilpotent Tortkara algebras

“…Another interesting direction in classifications of algebras is geometric classification. We refer the reader to [27][28][29]32] for results in this direction for Jordan, Lie, Leibniz, Zinbiel and other algebras. In the present paper, we give algebraic classification of nilpotent algebras of a new class of non-associative algebras introduced by Dzhumadildaev in [15].…”

Degenerations of Filippov algebras

“…There are also works in which the full information about degenerations was found for some variety of algebras. Here one can mention the descriptions of degenerations of low dimensional associative, Lie, pre-Lie, Malcev, Leibniz (see [7,24,25]) and all 2-dimensional (see [26]) algebras. There is an obvious connection between the notions of degeneration and deformation.…”

The geometric classification of nilpotent Tortkara algebras

“…Zinbiel algebras were introduced by Loday in [32] and studied in [12,13,15,16,28,33,35]. Under the Koszul duality, the operad of Zinbiel algebras is dual to the operad of Leibniz algebras.…”