TITLE: What is a (super)representation of a Lie group(oid)?

ABSTRACT:
Lie groupoids are a generalization of Lie groups that incorporate both "internal" and "external" symmetries. Representation theory is a valuable tool for studying Lie groups, so it would be nice to similarly utilize representation theory in the study of Lie groupoids. There is a reasonably natural definition of Lie groupoid representation, but it unfortunately fails to include an adjoint representation. However, it turns out that this problem can be avoided if we allow for a more general notion of a superrepresentation.

First, I will explain what a Lie groupoid is, give some concrete examples, and illustrate the problem one encounters in trying to define an adjoint representation. Next, I will describe how the definition of a representation of a Lie group can be restated in the language of homological algebra. In this point of view, the definition of superrepresentation is straightforward, and the result can be translated back into conventional language. In the case of Lie groupoids, we can define "adjoint superrepresentations", but they are not canonical. If there is time, I will describe the canonical object of which these adjoint superrepresentations are manifestations.