Equivariant map superalgebras

Details Equivariant map superalgebras Equivariant map superalgebras

2012-02-19

arxiv.org/abs/1202.4127v1

Mathematics

Representation Theory

27 pages

Scientific paper

Suppose a group Gamma acts on a scheme X and a Lie superalgebra g. The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from X to g. We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of X is finitely generated, Gamma is finite abelian and acts freely on the rational points of X, and g is a basic classical Lie superalgebra (or sl(n,n), n > 0, if Gamma is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on X. Furthermore, in the case that the even part of g is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of g is not semisimple (more generally, if g is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras.

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