A differential graded bialgebra associated to settheoretical solutions of the Yang-Baxter equation

JULIANA GARCÍA GALOFRE; MARCO FARINATI

Congreso; XX Coloquio Latinoamericanode A´lgebra ?O. Villamayor?; 2014

We define a bialgebra B such that it?s homology and cohomology equals bi-quandle homology and cohomology defined in [CJKS] and other generalizations ofcohomology of rack-quanlde case (for example defined in [CES]). This algebraicstructure allows us to show the existence of an associative product in the cohomol-ogy of biquandles This product was known for the rack case (with a topologicalproof, we provide a totally algebraic and independent proof) but unknown in gen-eral biquandle case.[CJKS]=S. Carter, D. Jelsovskyb, S. Kamada, M. Saito Quandle homology groups, theirBetti numbers, and virtual knots Journal of Pure and Applied Algebra 157 (2001)135155.[CES]=J.Scott Carter, Mohamed Elhamdadi and Masahico Saito. Homology Theory forthe Set-Theoretic Yang-Baxter Equation and Knot Invariants from Generalizationsof Quandles. Fund. Math., 184 (2004), 31-54 doi:10.4064/fm184-0-3