p -Filiform Zinbiel algebras

Abstract

The paper deals with the classification of a subclass of finite-dimensional Zinbiel algebras: the naturally graded p-filiform Zinbiel algebras. A Zinbiel algebra is the dual to Leibniz algebra in Koszul sense. We prove that there exists, up to isomorphism, only one family of naturally graded p-filiform Zinbiel algebras under hypothesis n − p ≥ 4.

Degenerations of Zinbiel and nilpotent Leibniz algebras

“…Under the Koszul duality the operad of Zinbiel algebras is dual to the operad of Leibniz algebras. Zinbiel algebras were studied in [1,2,13,16,17]. 1 The work was supported by RFBR 16-31-50017, 14-01-00014, FAPESP 14/19521-3, 14/24519-8 and by R & D 6.38.191.2014 of Saint-Petersburg State University, "Structure theory, classification, geometry, K-theory and arithmetics of algebraic groups and related structures".…”

Classification of Naturally Graded Zinbiel Algebras with Characteristic Sequence Equal to (n–p, p)

“…Applying the relations (7), (8) and arguments similar to the ones that are used in the proof of Proposition 3.1, we derive the relation…”

“…. , 1)) was obtained in [8], in order to complete the description similar to [11], in this paper we present the description (up to isomorphism) of naturally graded Zinbiel algebras with characteristic sequence equal to (n − p, p).…”

Central extensions of filiform Zinbiel algebras

“…Zinbiel algebras were introduced by Loday in [43] and studied in [2,5,10,17,[20][21][22]41,[44][45][46]49]. Under the Koszul duality, the operad of Zinbiel algebras is dual to the operad of Leibniz algebras.…”