Representation of Groupoids

Question

The title is vague, my actuall question is the following:

Has the representations of groupoids been systematically studied? Is there any new phenomenon, compare with the representation of groups? (Either brand new things or pitfalls for those who too familiar with representations of groups). Does this point of view simplified any proof of theorems in representation of groups?

I've been only think of this for 15 mins. But I feel like it might be helpful to think of representation of groupoids for the following reasons:

1. When one talk about local systems, thinking of it as "representation of the fundamental groupoid" seems more natural than talking about "representation of the fundamental group".

2. When we talk about modules on stacks, if we choose a presentation of the stack (which is a groupoid), can we treat a given module as a representation of the groupoid? (Just as modules on BG gives representations of G.) Studying when two representations give the "same" module will be interesting.

Answer 1

A representation of any categorical object (e.g. "groupoid" = "1-category with only isomoprhisms") is simply a (nice) functor from that object to the category of Vector Spaces. Then, as Chris points out, the abstract representation theory of groupoids essentially reduces to the representation theory of groups.

The story becomes much richer in the Lie category, because then you should ask for the representation to be smooth. The story has not been completely told, and even the parts that have been told I don't know well. For a hint at some of the interesting behavior, see https://arxiv.org/abs/0810.0066. I don't know anything about the algebraic category, but I believe that there's interesting stuff there too (as far as I can tell, algebraic stacks are more complicated than smooth ones).

Hopefully someone who knows the literature better than I can say more.

Answer 2

The "classical" definition of a representation of a Lie groupoid is rather similar to that of a Lie group. For a Lie group representation, you start with a vector space $V$ and define a representation to be a smooth homomorphism $G \to Gl(V)$. Given a Lie groupOID $G \rightrightarrows M$, to form a (classical) representation of $G$, you need to start not with a vector space $V$, but with a vector BUNDLE $V \to M$. Then, from $V$ you can construct a Lie groupoid $Gl(V) \rightrightarrows M$ (the arrows are linear isomorphisms between the fibres of V). A representation for $G$ is simply a Lie groupoid homomorphism $G \to Gl(V)$.

It should be noted that this notion of representation is somehow "too strict". Giorgio Trentinaglia argues that one should instead replace smooth vector bundles with more general objects, which he calls "smooth Euclidean fields". In this setting, he proves a version of Tannaka duality for proper Lie groupoids. You can read about this in his paper:

Tannaka duality for proper Lie groupoids, Journal of Pure and Applied Algebra.

There is also an arxiv version of this.

Even more, here is a link to his thesis which should provide even more detail:

http://igitur-archive.library.uu.nl/dissertations/2008-0904-200909/trentinaglia.pdf

Answer 3

Every groupoid is equivalent to a disjoint union of groups. In fact the inclusion of the sub-2-category of disjoint unions of groups into all groupoids is an equivalence. Hence the representation theory of groupoids reduces to that of groups.

Answer 4

People in operator theory study this a lot. Look at Jean Renault's Springer lecture notes on groupoid C*-algebras or the book by Alan Paterson: Groupoids, Inverse Semigroups and their operator algebras. Here they use Hilbert bundles to define representations.

I think in the finite case that groupoids give a nice take on induced representations. If you have a subgroup H of a group G, then G has a covering groupoid corresponding to H. It is the category of elements for the action of G on G/H. The covering groupoid is naturally equivalent to H and so has the same representation theory as H. Now the category of representations of the covering groupoid is the module category of the corresponding category algebra since there are finitely many vertices. There is a homomorphism from G into this category algebra that sends an element g to the sum of its preimages under the covering morphism. It is easy to see that if you start with a representation of H, take the corresponding representation of the covering groupoid, turn it into the representation of the category algebra of the groupoid and compose with the homomorphism from G above, you get the induced representation.

Answer 5

Groupoïds don't add much in theory but they do make some statements much simpler and they are sometimes necessary to get some intuition.

A typical example is the Van Kampen theorem. To state it in terms of fundamental groups $\pi_1(X,x)$ you have to chose one base point for each connected component. In terms of fundamental groupoïds though, it's elementary: $\Pi_1(X\cup_Z Y) = \Pi_1(X) *_{\Pi_1(Z)} \Pi_1(Y) $.

The reason the usual statement of the theorem is equivalent to this one is that any groupoïd is equivalent to a disjoint sum of groups. But equivalent does not mean equal; isomorphic does not mean canonicaly isomorphic. A statement about groupoïds translate into a statement about groups up to conjugacy and this kind of subtlety can get very tricky (and/or interesting) in practice.

Deligne's theory of the motivic unipotent fundamental group "Le groupe fondamental de la droite projective moins trois points" gives a great illustration of this fact. It gives a thoery of groupoïds and their represenations in fibered categories. It also shows that even in the elementary case of $P^1- \{ 0,1,\infty \}$ you have to work with fundamental groupoïds to really understand the arithmetic aspects of the fundamental groups.

Answer 6

There is a theory of representation of groupoids, that runs parallel to the classical theory of locally compact groups. A locally compact Hausdorff group $G$ always has a left (as well as a right) invariant measure. Using this measure one can define the convolution *-algebra of the group. A classical result is, that there is a natural bijection between the irreducible unitary representations of $G$ and non-degenerate *-representations of the convolution algebra of $G$.

On the similar lines, J. Renault proves, that if $G$ is a locally compact, Hausdorff groupoid with a left invariant continuous family of measures, then one can form a convolution *-algebra of $G$. An irreducible unitary representation of $G$ are defined using a measurable bundle of Hilbert spaces over the space of units of $G$ and a quasi-invariant measure on the space of units of $G$. Then he proves that there is a natural bijection between the irreducible unitary representations of $G$ and the convolution *-algebra of $G$.

When $G$ is a group, this theory is same as the classical theory representation theory.

Indeed, proving this theory for groupoids is very hard and there are many issues to take care of. Groupoids behave differently than groups. For example, a locally compact Hausdorff groupoid need not carry an invariant continuous family of measures. One has to demand the family of measures, as data. As mentioned earlier, a good reference for this purpose is Renault's book,``A groupoid approach to $C^*$-algebras. This theory is very useful for studying dynamical systems and associated $C^*$-algebras.