Averaging antisymmetric infinitesimal bialgebra and perm bialgebras
Bo Hou, Zhanpeng Cui

TL;DR
This paper develops a new bialgebra theory for averaging algebras, introducing averaging antisymmetric infinitesimal bialgebras, and explores their connections to Yang-Baxter solutions, Frobenius algebras, and perm bialgebras.
Contribution
It generalizes antisymmetric infinitesimal bialgebras to averaging algebras, introduces new concepts like $ ext{O}$-operators, and links these structures to perm bialgebras.
Findings
Characterization of averaging antisymmetric infinitesimal bialgebras via Frobenius algebras
Construction of solutions to the Yang-Baxter equation in averaging algebras
Extension of perm bialgebras from averaging algebra frameworks
Abstract
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 -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. Moreover, we introduce the notion of factorizable averaging antisymmetric infinitesimal bialgebras and show that a factorizable averaging antisymmetric…
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons
