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Mathematische Zeitschrift

, Volume 277, Issue 1–2, pp 373–399 | Cite as

Equivariant map superalgebras

  • Alistair Savage
Article

Abstract

Suppose a group \(\Gamma \) acts on a scheme \(X\) and a Lie superalgebra \(\mathfrak {g}\). The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from \(X\) to \(\mathfrak {g}\). 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 \(\mathfrak {g}\) is a basic classical Lie superalgebra (or \(\mathfrak {sl}\,(n,n)\), \(n \ge 1\), 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 \(\mathfrak {g}\) 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 \(\mathfrak {g}\) is not semisimple (more generally, if \(\mathfrak {g}\) 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.

Keywords

Lie superalgebra Basic classical Lie superalgebra Loop superalgebra Equivariant map superalgebra Finite dimensional representation  Finite dimensional module 

Mathematics Subject Classification (2010)

17B65 17B10 

Notes

Acknowledgments

The author would like to thank Shun-Jen Cheng, Daniel Daigle, Dimitar Grantcharov, Erhard Neher and Hadi Salmasian for helpful discussions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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