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The definition reads that "A G-extension of a fusion category D is a G-graded fusion category C whose trivial component is equivalent to D." It seems like a priori there can be multiple G-extensions for the same fusion category D. Is that really the case (i.e. no reduction mechanism)? But there seems to be a "canonical one" at least, which has the same category D sitting on top of each component of the grading. Does this particular G-extension have a name?

flag	asked Nov 19 at 0:33 Lucy Zhang 43●3

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If the words you're using all mean what I think they mean, then no, extensions aren't unique: you can find nonequivalent $G$-extensions of $\text{Vect}$, for example, by twisting the associator by a class in $H^3(G, \mathbb{C}^{\ast})$. – Qiaochu Yuan Nov 19 at 0:56

Thanks! This is great to know! Fits exactly. :-) – Lucy Zhang Dec 16 at 11:48

César Galindo · Accepted Answer · 2012-11-21 23:14:12Z UTC

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A complete reference for G-extension of fusion categories is http://arxiv.org/abs/0909.3140 (see also http://arxiv.org/abs/0911.0881). The kind of examples that you are saying are $\mathcal D\boxtimes \text{Vec}_G$, the Deligne product of $\mathcal D$ with Vec$_G$.

answered Nov 21 at 23:14

César Galindo
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Hey, Cesar! Thanks. Sorry for being off for a while. So, we can also twist the Deligne product, I assume. Let me look a bit more carefully into the reference paper. But it's always good to be able to just discuss intuitive. Hope to see you again soon. – Lucy Zhang Dec 16 at 11:10

Is the G-extension of a fusion category unique?

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