In particular this defines a subgroup [epsilon]V [subset] BrPic(H-mod) which is the homomorphic image of the group [Aut.sub.mon] (H-mod).
Our previous lemma applied to [chi] = DA-mod gives a Z(H-mod)-Z(r(H)-mod)-bimodule category M' = [chi], which is in general not invertible.
For these cases partial dualizations give rise to elements in BrPic(H-mod).
This can lead to the effect that H-mod [congruent to] H'-mod where H has a semi direct decomposition while H' has not, but still both centers carry the respective partial actualization.
The monoidal equivalences Rep(G') [right arrow] H-mod are given by Bigalois objects [.sub.f][R.sub.Rep(G')] where H is the Doi twist of C[G'] and f [member of] [Aut.sub.Hopf](H).
Already the well-known fact that BrPic(H-mod) [congruent to] [Aut.sub.br](DH-mod) has interesting implications for H = B (M) x C[G] as we have already seen in the Taft algebra case:
Note that this gives bimodule categories between categories H-mod and L-mod that are very different as categories.
From a physical perspective it very interesting to study such defects between different phases labeled H-mod and L-mod, in particular where H is the Borel part of a quantum group and L is a different lifting.
of the category H-mod by [Vect.sub.[SIGMA]] are associated to homomorphisms [psi] : [SIGMA] [right arrow] BrPic(C) (plus additional coherence data we omit here) with [D.sub.t] = [psi](t) a C-C-bimodule category.
We finally sketch briefly what the result is for C = H-mod when [psi] lands in our three subgroups BV, [epsilon]V, (R) in BrPic(H-mod).
Let [psi] : [SIGMA] [right arrow] BV = Ind([Aut.sub.mon](H-mod)).
Then D = H-mod where the new Hopf algebra is as an algebra