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.

In this work, the hom-center-symmetric algebras are constructed and discussed. Their bimodules, dual bimodules and matched pairs are defined. The relation between the dual bimodules of hom-center-symmetric algebras and the matched pairs of hom-Lie algebras is established. Furthermore, the Manin triple of hom-center-symmetric algebras is given. Finally, a theorem linking the matched pairs of hom-center-symmetric algebras, the hom-center-symmetric bialgebras and the matched pairs of sub-adjacent hom-Lie algebras is provided.

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