The Unapologetic Mathematician

Mathematics for the interested outsider

Category Representations

We’ve seen how group representations are special kinds of algebra representations. But even more general than that is the representation of a category.

A group is a special monoid, within which each element is invertible. And a monoid is just a category with a single object. Similarly, an \mathbb{F}-algebra is just like a monoid but enriched over the category of vector spaces over \mathbb{F}. That is, it’s a one-object category with an \mathbb{F}-bilinear composition. It makes sense to regard both of these structures as categories of sorts. A representation will then be a functor from one of these categories.

The clear target category is \mathbf{Vect}_\mathbb{F}. So what’s a functor \rho from, say, a group G (considered as a category) to \mathbf{Vect}_\mathbb{F}? First the single object of the category G picks out some object V\in\mathbf{Vect}_\mathbb{F}. That is, V is a vector space over \mathbb{F}. Then for each arrow g in G — each group element — we have an arrow \rho(g)\in\hom_\mathbb{F}(V,V). Since g has to be invertible, this \rho(g) must be invertible — an element of \mathrm{GL}(V).

What about an algebra? Now our source category A and our target category \mathbf{Vect}_\mathbb{F} are both enriched over \mathbf{Vect}_\mathbb{F}. It only makes sense, then, for us to consider \mathbb{F}-linear functors. Such a functor F again picks out a single vector space V for the single object of A (considered as a category). Every arrow a in A gets sent to an arrow \alpha(a)\in\hom_\mathbf{F}(V,V). This mapping is linear over the field \mathbb{F}.

So what do category representations get us? Well, one thing is this: consider a combinatorial graph — a collection of “vertices” with some directed “edges” joining them. A path in the graph is a sequence of directed edges joined tip-to-tail, and the collection of all paths in the graph constitutes the “path category” of the graph (exercise: identify the identity paths). A representation of this path category is what mathematicians call a “quiver representation”, and they’re big business.

More interesting to me is this: the category \mathcal{T}ang of tangles (or \mathcal{OT}ang of oriented tangles, \mathcal{F}r\mathcal{T}ang of framed tangles, or \mathcal{F}r\mathcal{OT}ang of framed, oriented tangles). This is a monoidal category with duals, as is \mathbf{Vect}_\mathbb{F}, and so it only makes sense to ask that our functors respect those structures as well. We don’t ask that it send the braiding to the symmetry on \mathbf{Vect}_\mathbb{F}, since that would trivialize the structure.

So what is a representation of the category \mathcal{T}ang? It is my contention that this is nothing but a knot invariant, viewed in a more natural habitat. A little more generally, knot invariants are the restrictions to knots (and links) of functors defined on the category of tangles, which can often (always?) be decategorified — or otherwise rendered down — into representations of \mathcal{T}ang. This is my work: to translate existing knot theoretical ideas into this algebraic language, where I believe they find a better home.

October 27, 2008 Posted by | Algebra, Category theory, Linear Algebra, Representation Theory | 7 Comments