Equivariant map algebras                                                                                                                               
                 -- Alistair Savage, February 19, 2010  

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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.