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Equivariant map superalgebras

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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.

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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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Correspondence to Alistair Savage.

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This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

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Savage, A. Equivariant map superalgebras. Math. Z. 277, 373–399 (2014). https://doi.org/10.1007/s00209-013-1261-7

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