A differential bialgebra associated to a set theoretical solution of the Yang-Baxter equation

TL;DR

This paper introduces a differential graded bialgebra associated with set-theoretical solutions of the Yang-Baxter equation, connecting it to biquandle homology and cohomology, and establishing an associative product in biquandle cohomology.

Contribution

It constructs a new algebraic structure linking biquandle homology to Hochschild (co)homology, enabling novel algebraic and homological insights.

Findings

Defined a dg bialgebra for set-theoretical Yang-Baxter solutions.

Established an associative product in biquandle cohomology.

Connected biquandle (co)homology with Hochschild (co)homology.

Abstract

For a set theoretical solution of the Yang-Baxter equation $(X,\sigma)$, we define a d.g. bialgebra $B=B(X,\sigma)$, containing the semigroup algebra $A=k\{X\}/\langle xy=zt : \sigma(x,y)=(z,t)\rangle$, such that $k\otimes_A B\otimes_Ak$ and $\mathrm{Hom}_{A-A}(B,k)$ are respectively the homology and cohomology complexes computing biquandle homology and cohomology defined in \cite{CJKS} and other generalizations of cohomology of rack-quanlde case (for example defined in \cite{CES}). This algebraic structure allow us to show the existence of an associative product in the cohomology of biquandles, and a comparison map with Hochschild (co)homology of the algebra $A$.