Bigalois Groupoid Of Drinfel'd Group Double

Question

2-cocycles of a given Hopf algebras $H$ no longer form a group, but a groupoid between different Doi twists of the Hopf algebra $H,L$. The subgroup of "lazy" 2-cocycles precisely preserve the underlying Hopf algebra. They are usually presented as $H$-$L$- resp. $H$-$H$-Bigalois Objects.

Now we know from Schauenburg, that grouprings $H$ by cocommutativity have only lazy 2-cocycles (those of the group) and $Bigal(H)=Aut(G)\ltimes H^2(G,k^*)$

We also know the structure of lazy 2-cocycles (and resp. Bigalois objects) on tensor products (Schauenburg) and Drinfel'd doubles (Bichon,Carnovale) in terms of the smaller Hop algebra one's and the pairings between them....

...but NOT the respective larger entire Bigalois grouppoid....

...but I want FAR LESS: namely just the Bigalois grouppoid on the Drinfel'd double of a group! I searched the web, but could note even find a description of the Doi twists...

Any idea or hint would be greatly appreciated! Thanx in advance

Answer 1

For a finite dimensional Hopf algebra H, $Bigal(D(H))=Bigal(H\otimes H^*)$, becuase $^{D(H)}\mathcal{M}\cong_{\otimes}\ ^{H\otimes H^*}\mathcal{M}$.

If $H=kG$ (Gfinite group $G$), then you need first the description of all Bigalois objects of $\mathcal{O}_k(G)$. A description using other name was done by Davydov in http://arxiv.org/abs/0708.2758 (under the restriction that 2 does not divide $|G|$). Using that description and Shauenburg's result about abut Galois objects on tensor products you can describe all Bigalois Objects of the Drinfeld double of a finite group.