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 [Formula: see text]-invariant and [Formula: see text]-symmetric. The relationship between AYBPs in symmetric Frobenius algebras and Rota–Baxter systems is investigated using the formalism of [Formula: see text]-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.