Quasi-triangular, factorizable anti-symmetric covariant bialgebras and symmetric Rota-Baxter Frobenius algebras

This paper introduces the notion of an anti-symmetric covariant (ASC) bialgebra as a generalization of a balanced infinitesimal bialgebra. Our primary focus is on the construction of these algebraic structures from associative Yang-Baxter pairs (AYBPs). We demonstrate that this construction leads to a quasi-triangular ASC bialgebra under the conditions that the AYBP is τ-invariant and τ-symmetric. The relationship between AYBPs in symmetric Frobenius algebras and Rota-Baxter systems is investigated using the formalism of O-operator systems. A central theme of this work is the development of a factorization theory for a special class of these structures, termed factorizable ASC bialgebras. We prove that a factorizable ASC bialgebra induces a factorization of its underlying associative algebra. To characterize these structures, we introduce the concept of a symmetric Rota-Baxter Frobenius algebra and establish a one-to-one correspondence between them and factorizable ASC bialgebras. This correspondence is subsequently employed to construct examples of factorizable ASC bialgebras from Rota-Baxter systems. Finally, we present an algorithm for computing AYBPs in finite-dimensional associative algebras and apply it to classify all such pairs in two-dimensional complex associative algebras, yielding concrete examples of the aforementioned structures.