Cohomology and crossed products for weak Hopf algebras

Ramon Gonzalez Rodriguez
Friday, July 26, 2013 - from 17:00 to 17:25
In [6] Sweedler introduced the cohomology of a cocommutative Hopf algebra $H$ with coefficients in a commutative $H$-module algebra $A$ and he gave an interesting interpretation of the second group of this cohomology, denoted by $H_{\varphi_{A}}^2(H,A)$, in terms of equivalence classes of cleft extensions, i.e., classes of equivalent crossed products determined by a 2-cocycle. This result was extended by Doi [4] proving that, in the non commutative case, there exists a bijection between the isomorphism classes of $H$-cleft extensions $B$ of $A$ and equivalence classes of crossed systems for $H$ over $B$. If $H$ is cocommutative the equivalence is described by $H_{\varphi_{\mathcal{Z}(B)}}^{2}(H,\mathcal{Z}(B))$ where $\mathcal{Z}(B)$ is the center of $B$. With the recent arise of weak Hopf algebras (quantum groupoids), introduced by B\"ohm, Nill and Szlach\'anyi in [3], the notion of crossed product can be adapted to the weak setting. The key to extend the Hopf crossed product constructions to the weak world can be found in [5] where general product was defined for an algebra $A$ and an object $V$ both living in a strict monoidal category $\mathcal C$ where every idempotent splits. Then, if in the Hopf algebra setting the second cohomology group classify crossed products of $H$ and a commutative left $H$-module algebra $A$, what about the weak seetting? The answer to this question is the main motivation of this talk. More precisely, we show that if $H$ is a cocommutative weak Hopf algebra and $A$ is a commutative left $H$-module algebra, all the weak crossed products defined in $A\otimes H$ with a common preunit can be described by the second cohomology group of a new cohomology, that we call the Sweedler cohomology of a weak Hopf algebra with coefficients in $A$. The results that will be presented are part of a joint work with J.N. Alonso and J.M. Fern\'andez (see [1] and [2]).