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. After the seminar we will take the speaker to lunch. See the Algebra Seminar web page for information about other seminars in the series. |