Anticenter-Symmetric Bialgebras

This paper develops a bialgebra theory for anticenter-symmetric algebras by introducing the concept of an anticenter-symmetric bialgebra, equivalent to a Manin triple of anticenter-symmetric algebras. A study of this framework leads to the anticenter-symmetric Yang-Baxter equation in anticenter-symmetric algebras, analogous to the classical Yang-Baxter equation in Mock Lie algebras and the associative Yang-Baxter equation.An unexpected finding is that the anticenter-symmetric and associative Yang-Baxter equations share the same form. Additionally, skew-symmetric solutions to the anticenter-symmetric Yang-Baxter equation define anticenter-symmetric bialgebras. To advance the theory, the paper introduces O-operators and pre-anticentersymmetric algebras, which facilitate the construction of these solutions and provide a foundation for further exploration.