Artin-Wedderburn in fusion categories

In quantum algebra we’re often studying some classical algebraic notion, but instead of working in the category of vector spaces you instead work in a more general tensor category. For example, the theory of finite type knot invariants is roughly the theory of simple Lie algebra objects in symmetric tensor categories, while the theory of subfactors is roughly that of simple algebra objects in unitary tensor categories. The basic question is then which notions from the classical theory generalize to the quantum setting. For example, is there an analogue of Artin-Wedderburn for semisimple algebra objects in fusion categories? The goal of this post is to argue that the following theorem (due to Ostrik, modulo any errors I’ve introduced) gives a satisfactory generalization.

Any semisimple algebra object in a fusion category \mathscr{C} is isomorphic (as an algebra object) to the internal endomorphisms End(X) for some object X in a semisimple module category \mathscr{M} over \mathscr{C}.

First I’ll unpack the definitions in this statement and then I’ll explain how Artin-Wedderburn for semisimple algebras over a fixed field k follows from this statement. I’ve been thinking about this theorem because Pinhas Grossman and I have been using it to classify “quantum subgroups” of the Haagerup fusion categories, but that’s a story for another day.

First a fusion category over a field k is a k-linear semisimple category with finitely many simple objects which has a tensor product, duals, and a trivial object. An algebra object A in \mathscr{C} is an object together with multiplication and unit morphisms satisfying the usual relations. Given an algebra A in \mathscr{C} you can look at the A-module objects in \mathscr{C}, which are pairs an object M in \mathscr{C} and a morphism A \otimes M \rightarrow M satisfying associativity. We call A semisimple if the category of A-module objects is itself semisimple.

A tensor category is a “categorification” of a ring, so just like you need modules to understand ring theory you should “categorify” the notion of module to study tensor categories. This suggestions the idea of a “module category” which is a category on which the tensor category acts in an appropriate sense. The prototypical example of a module category is the category of A-modules, but there’s a slightly annoying technicality: the category of left A-modules is a right module category (and vice-versa). For another example, if you have a forgetful functor from \mathscr{C} to vector spaces that turns Vec into a module category over \mathscr{C}. Similarly, if H is a subgroup of G then Rep(H) is a module category over Rep(G) with the action given by restrict and then tensor product.

Finally taking “internal endomorphisms” is a process which takes an object X in a module category \mathscr{M} and returns an algebra object in \mathscr{C} which “behaves like End(X) in an appropriate sense. More precisely, it is the unique object (up to unique isomorphism) which represents the functor C \mapsto Hom(C \otimes X, X).

You can also see an old blog post of Urs’s on this topic, as well as Victor’s paper mentioned above.

Ok, now that I’ve explained the theorem why is it useful and why does it have anything to do with Artin-Wedderburn? Well, the basic idea is that all module categories are a direct sum of indecomposable module categories, and all objects are direct sums of simple objects, hence once you have all the indecomposable module categories in hand you immediately get the list of all objects in all fusion categories and thus the list of all semisimple algebra objects.

Let’s work this out for the category of k vector spaces Vec_k. Since Vec_k only has one simple object, any indecomposable semisimple module category over Vec_k must only have one simple object. Thus the module category is determined by the ring of endomorphisms of this unique object, which must be a division ring over k. Hence, any module category is a direct sum \bigoplus_i Vec_{D_i} and any object in such a module category is a direct sum \bigoplus_i X_i^{\oplus n_i} where X_i is the unique simple object in Vec_{D_i}. A simple calculation then shows that the internal endomorphisms of this object X is thus \bigoplus_i M_{n_i}(D_i) which is exactly what Artin-Wedderburn says!

As a final note, you might object that Artin-Wedderburn is really a statement about semisimple rings, not semisimple algebras over a fixed field. There is again an analogue of this Artin-Wedderburn for rings using some results of Kuperberg.

2 thoughts on “Artin-Wedderburn in fusion categories

  1. I think the gremlins got at your definition of the the endomorphism object. If X is an object of your module category \mathscr{M} then I guess you want End(X) to be the object of \mathscr{C} which satisfies, for any C in \mathscr{C},

    Hom_{\mathscr{M}}(C\otimes X,X)\cong Hom_{\mathscr{C}}(C,End(X)).

    Is this really called an `internal’ endomorphism object for $X$? I would expect a thing so-named to live in \mathscr{M} rather than in \mathscr{C}.

  2. Indeed, I’d copy-pasted out of the paper and forgot to fix the notation. Thanks!

    That’s a good point about expecting “internal hom” to live in \mathscr{M}, but nonetheless this is the standard name. I think the idea is that it generalizes the usual internal Hom in the case where \mathscr{M} is the “regular representation” (that is \mathscr{M} = \mathscr{C} with the action given by the tensor product in \mathscr{C}.

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