Speaker:

Ramon Gonzalez Rodriguez

Friday, July 26, 2013 - from 17:00
to 17:25

Abstract:

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]).