Center-symmetric algebras and bialgebras: Relevant properties and consequences

Abstract

Lie admissible algebra structures, called center-symmetric algebras, are defined. Their 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 the 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 triples of center-symmetric algebras are 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 triples of center-symmetric algebras, matched pairs of Lie algebras and center-symmetric bialgebras is provided.