MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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 '12 at 0:56
Thanks! This is great to know! Fits exactly. :-) – Lucy Zhang Dec 16 '12 at 11:48
up vote 2 down vote accepted

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

share|cite|improve this answer
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 '12 at 11:10

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.