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

摘要

For a set theoretical solution of the Yang-Baxter equation $(X,\sigma)$, wedefine 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 $k\otimes_AB\otimes_Ak$ and $\mathrm{Hom}_{A-A}(B,k)$ are respectively the homology andcohomology complexes computing biquandle homology and cohomology defined in\cite{CJKS} and other generalizations of cohomology of rack-quanlde case (forexample defined in \cite{CES}). This algebraic structure allow us to show theexistence of an associative product in the cohomology of biquandles, and acomparison map with Hochschild (co)homology of the algebra $A$.