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), 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 circle times(A) B circle times(A) k and Hom(A-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. (C) 2016 Elsevier B.V. All rights reserved.