## University of Sydney Algebra Seminar

# Alistair Savage (University of Ottawa)

## Wednesday 12th May, 12.05-12.55pm, Carslaw 159 (note unusual day and venue)

### Equivariant map algebras and their representation theory

Suppose a finite group acts on a scheme (or algebraic variety) *X* and a
finite-dimensional Lie algebra *g*. Then the space of equivariant
algebraic maps from *X* to *g* is a Lie algebra under pointwise
multiplication. Examples of such equivariant map algebras include
(multi)current algebras, (multi)loop algebras, three point Lie
algebras, and the (generalized) Onsager algebra. In this talk we will
present a classification of the irreducible finite-dimensional
representations of an arbitrary equivariant map algebra. It turns out
that (almost) all irreducible finite-dimensional representations are
evaluation representations. As a corollary, we recover known results
on the representation theory of particular equivariant map algebras
(for instance, the loop algebras and the Onsager algebra) as well as
previously unknown classifications of other equivariant map algebras
(for example, the generalized Onsager algebra). All such
classifications are specializations of the general theorem. This is
joint work with Erhard Neher and Prasad Senesi. Time permitting, we
will discuss some further ongoing work with E. Neher on a related
subject.