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

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 $A=k\{X\}/⟨xy=zt:\sigma (x,y)=(z,t)⟩$, such that $k{\otimes }_{A}B{\otimes }_{A}k$ 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 $A$.

arXiv comment23 pages, proof of Theorem 5 written with more details, some references added