Central extensions of filiform Zinbiel algebras

Abstract

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an n-dimensional nullfiliform Zinbiel algebra is isomorphic to an (n + 1)-dimensional null-filiform Zinbiel algebra. Moreover, we obtain all pairwise non isomorphic quasi-filiform Zinbiel algebras.

Zinbiel algebras are nilpotent

“…Some properties of Zinbiel algebras were studied in [1,5,6]. Filiform Zinbiel algebras were described and classified in [1,3,4]. The classification of complex Zinbiel algebras up to dimension 4 was obtained in [6] and [13].…”

Abelian subalgebras and ideals of maximal dimension in Zinbiel algebras

“…More concretely, in [12] the authors proved that every finite-dimensional Zinbiel algebra over an algebraically closed field is solvable and it is nilpotent over the complex number field. Filiform Zinbiel algebras were described and classified in [1,7,8]. The classification of complex Zinbiel algebras up to dimension 4 was obtained in [12] and [23].…”

Unified products for Zinbiel 2-algebras

“…It has been appeared some interesting applications of Zinbiel algebras in multiple zeta values and construction of a Cartesian differential category [13,22]. For more recent studies of Zinbiel algebras, see [14,15,18].…”