MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

Following the notation of Etingof-Nikshych-Ostrik what is Out(G-mod) for a finite group G?

That is what are all bimodule cateogries over the fusion category G-mod of complex G-modules which have the property that they're just G-mod as a left (resp. right) G-module up to equivalence of bimodules? I think this is the same as the group of tensor autoequivalences of G-mod which do not come from conjugating by 1-dimensional objects, but ENO only prove that in the case where there are no 1-dimensional objects.

If you have an automorphism of the group G then you get an automorphism of G-mod. I'm not totally sure though if outer automorphisms of G necessarily give outer automorphisms (for example could you have an outer automorphism that fixed all conjugacy classes?) and I also don't know whether all outer automorphisms come up this way.

The reason that I'm asking is that I want to understand the relationship between Out(C) and Out(D) where C and D are Morita equivalent fusion categories. However, I realized I don't have enough examples where I understand what Out(C) actually is.

share|cite|improve this question
Nontrivial outer automorphisms can fix all conjugacy classes. See the 1947 paper by G.E. Wall ( for examples. Wall says that such examples were first given by Burnside in 1912, but that paper is apparently too old to appear on MathSciNet. – Evan Jenkins Jul 15 '10 at 23:24
up vote 3 down vote accepted

The group $\operatorname{Out}(G\operatorname{-Mod})$ is equivalent to the group $\operatorname{Aut}_{\textbf{Tw}}(k[G])$ of gauge equivalence classes of twisted automorphisms of the Hopf algebra $k[G]$ defined by Davydov in Twisted automorphisms of Hopf algebras. In Davydov's other paper, Twisted automorphisms of group algebras, he describes this group in the case that $\lvert G \rvert$ is relatively prime to 6. (The proposed description I gave in the earlier version of this answer fails in general because $\operatorname{Out}(G)$ is not in general a normal subgroup of $\operatorname{Aut}_{\textbf{Tw}}(k[G])$.)

First, we note that since $G\operatorname{-Mod}$ is symmetric, any inner tensor autoequivalence is automatically tensor equivalent to the identity, so it suffices to describe the group of isomorphism classes of tensor autoequivalences. Let $\omega: G\operatorname{-Mod} \to \operatorname{Vect}$ be the usual fiber functor. Any tensor autoequivalence $F: G\operatorname{-Mod} \stackrel{\cong}{\to} G\operatorname{-Mod}$ gives a new (tensor) fiber functor $\omega \circ F: G\operatorname{-Mod} \to \operatorname{Vect}$. This gives $G\operatorname{-Mod}$ the structure of the category of representations of some Hopf algebra; it is shown in the paper On families of triangular Hopf algebras by Etingof and Gelaki that this Hopf algebra will be a twist of the original one, and thus $F$ induces a twisted automorphism of $H$. Conversely, any twisted automorphism of $F$ gives rise to an autoequivalence of $G\operatorname{-Mod}$. Natural transformations of monoidal functors correspond to gauge equivalences of twisted automorphisms. Thus, we can identify isomorphism classes of monoidal autoequivalences of $G-\operatorname{Mod}$ with gauge equivalence classes of twisted automorphisms of $k[G]$.

share|cite|improve this answer

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.