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, σ), we define a d.g. bialgebra B = B(X, σ), containing the semigroup algebra A = k{X}/xy = zt : σ(x, y) = (z,t) , such that k ⊗A B ⊗A k and HomA−A(B, k) are respectively the homology and cohomology complexes computing biquandle homology and cohomology defined in [2,5] and other generalizations of cohomology of rack-quandle case (for example defined in [4]). This algebraic structure allows 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.