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If I don’t replace H by a fusion category but just by a ℂ-algebra, does that algebra have a name? It’s essentially “G-commutative” but I’m wondering if there’s an established name.
There is a notion of “crossed G-algebras”.
Ah, of course, thanks.
On a different note. Usually, when we talk about crossed modules, we think of group exact sequences 1→K→A→G→Γ→1 where (A→G) is the crossed module. In particular, if A→G is surjective, then Γ=1, and this crossed module is equivalent to (K→1). Now, thinking of A as a fusion category, K would be A1 the fusion subcategory of degree 1∈Γ. Moreover, if the Γ-grading is faithful, then this map A→G is surjective, which somewhat hints that this G-crossed braided fusion category A is equivalent to the braided fusion category A1. In what sense is this equivalence true?
I don’t know, haven’t really thought about it. But it sounds like there should be an evident comparison functor, in which case you could explicitly check whether it’s essentially surjective and fully faithful.
(rolled back an edit, I overlooked and repeated a pointer)
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