Conference abstracts

Session S03 - Hopf Algebras

July 25, 18:30 ~ 18:55

## A d.g. Hopf algebra associated to a set theoretical solution of the Yang-Baxter equation and cohomology

### Marco Farinati

### Uniersidad de Buenos Aires, Argentina - mfarinat@dm.uba.ar

For a set theoretical solution of the Yang-Baxter equation $(X,r)$, we define a d.g. Hopf algebra $B=B(X,r)$ containing the group algebra $k[G]$, where $G= \langle x\in X: xy=zt$ if $r(x,y)=(z,t)\rangle$, in such a way that $k\otimes_G B\otimes_G k$ and Hom${}_{G-G}(B,k)$ are respectively the homology and cohomology complexes computing quandle/rack homology and cohomology, as defined by knot theorists (Carter, Saito, Jelsovskyb, ElHamadi) and other generalizations of cohomology (e.g. twisted rack cohomology, or Yang-Baxter cohomology). This algebraic structure allow us to show the existence of an associative product in Yang-Baxter cohomology, and a comparison map with Hochschild (co)homology of k[G], that factors trough the Nichols algebra associated to $(X,-r)$.

Joint work with Juliana Garcia Galofre (Universidad de Buenos Aires, Argentina)..