Quasi-triangular and factorizable antisymmetric infinitesimal bialgebras

Abstract

In this paper, we introduce the notions of quasi-triangular and factorizable antisymmetric infinitesimal bialgebras. A factorizable antisymmetric infinitesimal bialgebra leads to a factorization of the underlying associative algebra. We show that the associative double of an antisymmetric infinitesimal bialgebra naturally enjoys a factorizable antisymmetric infinitesimal bialgebra structure. Then we introduce the notion of symmetric Rota-Baxter Frobenius algebras, and show that there is a one-to-one correspondence between factorizable antisymmetric infinitesimal bialgebras and symmetric Rota-Baxter Frobenius algebras. Finally, we show that a symmetric Rota-Baxter Frobenius algebra can give rise to an isomorphism from the regular representation to the coregular representation of a Rota-Baxter associative algebra.