Recall that the Brauer group of a field k consists of central simple algebras over k up to Morita equivalence with the group operation given by tensor product. For example, the Brauer group of the real numbers is Z/2 because the only central simple algebras are matrix algebras over or matrix algebras over the quaternions , and . It is a well-known and fundamental fact that the Brauer group is isomorphic to the second Galois cohomology where is the seperable closure of k.
What I’d like to explain in this post is a follow-your-nose proof of this isomorphism which comes from thinking about fusion categories. Namely, attached to any fusion category there is a very natural object called Brauer-Picard groupoid (introduced by Etingof-Nikshych-Ostrik). For the special case of the fusion category of vector spaces over k the Brauer-Picard groupoid has a point for every seperable extension of k and the group of automorphisms of the point k gives exactly the Brauer group. However, one can also look at the group of automorphisms of other points, in particular the point . The group of automorphisms of that point is instead naturally isomorphic to the Galois cohomology . Since the groupoid is connected we see that the Brauer group coincides with the Galois cohomology. In fact, there’s a natural choice of arrow from to and so a natural choice of isomorphism between the two groups.
This example came up in work in progress with Pinhas Grossman where we compute the Brauer-Picard groupoid of the fusion categories which come from the Haagerup subfactor. As we’ll see the automorphism group of a point in the Brauer-Picard groupoid has a subgroup consisting of certain “outer automorphisms.” I wanted to have a good example in hand where the outer automorphism group of different points were different in order to test certain lemmas. The example in this post is as extreme as things can get, for there are no nontrivial outer automorphisms, while for the whole group consists of outer automorphisms.
A fusion category over a field k is a k-linear rigid monoidal category (that is the hom spaces are k vector spaces, and there’s a good notion of duals and tensor products) which is semisimple and has finitely many isomorphism classes of simple objects and whose unit object is simple. You should think of a fusion category as like the category of representations of a finite group. In addition to the category of representations of a finite group, another good example to have in mind is the category of G-graded vector spaces.
Fusion categories are categorifications of rings (since their Grothendieck group has both an addition via direct sum and a multiplication via tensor product). Since it’s useful to study modules over rings, it stands to reason that one should also think about module categories over fusion categories. See Victor Ostrik’s paper for the details. The definition is about what you’d expect if you’re steeped in the right higher category theoretic notions of not being evil (i.e. composition need not associate on the nose because requiring equality is evil). In particular, it’s very natural to consider (higher) Morita equivalences of fusion categories. That is the invertible bimodule categories over two fusion categories. (Such Morita equivalences are interesting in their own right, but they’re also very intimately related to subfactors by the work of Mueger. The “standard invariant of a subfactor” is exactly a Morita equivalence of two unitary tensor categories.)
Now we’re ready to define the Brauer-Picard groupoid of a fusion category C. Its points are the fusion categories which are Morita equivalent to C, and its arrows are the Morita equivalences (up to equivalence of bimodule categories, otherwise you’d get some sort of 2-groupoid).
In order to calculate the Brauer-Picard groupoid of the category of k vector spaces, and thereby prove the isomorphism between Brauer group and Galois cohomology we need a couple key results of Victor Ostrik’s. First, any left module category over C can be realized as the category of right module objects over A for some algebra object A in C. (Yes that’s left module categories consist of right A-modules, that makes sense because when you look at right A-modules you can no longer act easily on the right only on the left.) Second, if M is a simple left C-module category then M gives a Morita equivalence between C and something called its “dual over M” which automatically acts on M on the right. In fact, a Morita equivalence between C and D is the same thing as a choice of module category M over C and a choice of isomorphism of C* with M (up to a certain equivalence). That is to say, in order to find all Morita equivalences between C and D we should find all module categories over C whose duals are isomorphic to D, and we should find all outer automorphisms of D (an automorphism is outer if it does not come from conjugation by an invertible object, modifying a bimodule category by an inner automorphism gives an isomorphic bimodule category).
Ok, now we’re ready. First let’s consider the point k. The module categories over k-Vect are all of the form mod-A for some algebra object A in k-Vect. The dual over this module category consists of exactly the A-A bimodules in k-Vect. Since the dual over the module category is supposed to have simple trivial object, we must have that A is simple. Furthermore, it is not difficult to see that A-mod-A is isomorphic to k-Vect exactly when A is central. Also Morita equivalent algebras yield the same module category. Finally it is not difficult to see that composition in the Brauer-Picard groupoid is given by tensor product. Hence the group of automorphism of k is exactly the Brauer group. (This fact is pointed out by ENO and motivated the “Brauer” part of the name of the Brauer-Picard groupoid.)
So what are the other points in this groupoid? Well we need to look at what the other fusion categories are that occur as A-mod-A for some k-algebra A. It is not hard to see that two such fusion categories are isomorphic (really I should say “equivalent” since categories are never isomorphic, but then you’d get confused about “equivalent” vs. “Morita equivalent”) exactly when they have the same center. Hence the points in the Brauer-Picard groupoid are exactly the field extensions of k.
What are the Morita equivalences between k-Vect and K-mod-K for some field extension K? Well first we want to classify the module categories over k-Vect and then we want to understand all outer automorphisms of K-mod-K. The former correspond to simple algebras over k whose center is K. In particular, when there’s only one. Hence all Morita autoequivalences of -mod- come from outer automorphisms. Now we’re almost done because as a rule of thumb outer automorphisms look like an appropriate second cohomology (in particular the outer automorphisms of G graded vector spaces are a semidirect product of Out(G) and H^2(G, k*).
The objects in -mod- correspond exactly to the Galois automorphisms . By definition, an outer automorphism of a monoidal category is a functor F: C->C together with a binatural transformation satisfying certain properties. In our setting you can show that the underlying functor must be the identity, and then it is immediate from the definition that the binatural transformation correspond exactly to elements of the second Galois cohomology. (In slightly more detail, to every pair of object (= elements of the Galois group) you assign an automorphism of (= scalar) satisfying certain coherence condtions (= cocycle) modulo the inner automorphisms (=coboundaries).)
You might wonder what happens when you look at another point K. Then the group of Morita autoequivalences is an extension of by Br(K).
(The astute reader will worry that when is infinite then one needs to be a bit more careful than I have been. Since this is a blog post and not a paper, I’m not going to try to sort those issues out here.)