Averaging antisymmetric infinitesimal bialgebras and induced perm bialgebras

In this paper, we establish a bialgebra theory for averaging algebras, called averaging antisymmetric infinitesimal bialgebras by generalizing the study of antisymmetric infinitesimal bialgebras to the context of averaging algebras. They are characterized by double constructions of averaging Frobenius algebras as well as matched pairs of averaging algebras. Antisymmetric solutions of the Yang–Baxter equation in averaging algebras provide averaging antisymmetric infinitesimal bialgebras. The notions of an [Formula: see text]-operator of an averaging algebra and an averaging dendriform algebra are introduced to construct antisymmetric solutions of the Yang–Baxter equation in an averaging algebra and hence averaging antisymmetric infinitesimal bialgebras. Finally, we apply the study of averaging antisymmetric infinitesimal bialgebras to perm bialgebras, extending the construction of perm algebras from commutative averaging algebras to the context of bialgebras, which is consistent with the well-constructed theory of perm bialgebras.