Irreducible finite-dimensional representations of equivariant map algebras

Details Irreducible finite-dimensional representations of equivariant map algebras Irreducible finite-dimensional representations of equivariant map algebras

2009-06-29

arxiv.org/abs/0906.5189v4

Trans. Amer. Math. Soc., 364 (2012), no. 5, 2619-2646

Mathematics

Representation Theory

25 pages; v2: results generalized to schemes and arbitrary finite-dimensional g; v3: change of notation, minor typos corrected

Scientific paper

10.1090/S0002-9947-2011-05420-6

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The corresponding equivariant map algebra is the Lie algebra M of equivariant regular maps from X to g. We classify the irreducible finite-dimensional representations of these algebras. In particular, we show that all such representations are tensor products of evaluation representations and one-dimensional representations, and we establish conditions ensuring that they are all evaluation representations. For example, this is always the case if M is perfect. Our results can be applied to multiloop algebras, current algebras, the Onsager algebra, and the tetrahedron algebra. Doing so, we easily recover the known classifications of irreducible finite-dimensional representations of these algebras. Moreover, we obtain previously unknown classifications of irreducible finite-dimensional representations of other types of equivariant map algebras, such as the generalized Onsager algebra.

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