Zinbiel algebras and bialgebras: main properties and related algebraic structures

TL;DR

This paper explores the fundamental properties of Zinbiel algebras and coalgebras, establishing their identities, bimodules, and bialgebra structures, and relating them to Lie and coassociative algebraic frameworks.

Contribution

It provides a comprehensive characterization of Zinbiel algebras and coalgebras, introduces their bialgebra structures, and links them to Lie and coassociative algebraic structures.

Findings

Zinbiel algebras are center-symmetric and Lie-admissible.

Construction of bimodules and matched pairs for Zinbiel algebras.

Definition of Zinbiel bialgebras via Manin triples.

Abstract

This work provides a characterization of left and right Zinbiel algebras.Basic identities are established and discussed, showing that Zinbiel algebras are center-symmetric, and therefore Lie-admissible algebras. Their bimodules are given, and used to build a Zinbiel algebra structure on the direct sum of the underlying vector space and a finite-dimensional vector space. In addition, their matched pair is built, and related to the matched pair of their sub-adjacent Lie algebras. Besides, Zinbiel coalgebras are introduced, and linked to their underlying Lie coalgebras and coassociative coalgebras. Moreover, the related Manin triple is defined, and used to characterize Zinbiel bialgebras, and their equivalence to the associated matched pair.