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

FARINATI, MARCO ANDRÉS; GARCIA GALOFRE, JULIANA

JOURNAL OF PURE AND APPLIED ALGEBRA

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2016 vol. 220 p. 3454 - 3475

0022-4049

For a set theoretical solution of the Yang-Baxter equation (X,r), we define a d.g. bialgebra B=B(X,r), containing the semigroup algebra A=k{X}/⟨xy=zt:r(x,y)=(z,t)⟩, such that k⊗AB⊗Ak and HomA−A(B,k) are respectively the homology and cohomology complexes computing biquandle homology and cohomology, and other generalizations of cohomology of rack-quanlde case. 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.