Center-symmetric algebras and bialgebras: relevant properties and consequences

TL;DR

This paper introduces and explores the properties of center-symmetric algebras and bialgebras, establishing their structural relationships and equivalences with Lie algebras, Manin triples, and matched pairs.

Contribution

It defines center-symmetric algebras and bialgebras, and establishes their connections with Lie algebras, Manin triples, and matched pairs, providing new theoretical insights.

Findings

Defined center-symmetric algebras and bimodules

Established the relation between Manin triples and matched pairs

Proved the equivalence between Manin triples, matched pairs, and bialgebras

Abstract

Lie admissible algebra structures, called center-symmetric algebras, are defined. Main properties and algebraic consequences are derived and discussed. Bimodules are given and used to build a center-symmetric algebra on the direct sum of underlying vector space and a finite dimensional vector space. Then, the matched pair of center-symmetric algebras is established and related to the matched pair of sub-adjacent Lie algebras. Besides, Manin triple of center-symmetric algebras is defined and linked with their associated matched pairs. Further, center-symmetric bialgebras of center-symmetric algebras are investigated and discussed. Finally, a theorem yielding the equivalence between Manin triple of center-symmetric algebras, matched pairs of Lie algebras and center-symmetric bialgebras is provided.