Irreducible finite-dimensional representations of equivariant map algebras

By E. Neher, Alistair Savage, Prasad Senesi. 2009

84 citations recorded by the source corpus.

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 particu- lar, 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 classifica- tions of irreducible finite-dimensional representations of other types of equivariant map algebras, such as the generalized Onsager algebra.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/17a92e2553e1f1f2788f84826ae89378420320d1

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In Algebra and Number Theory

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Citations in the corpus

BGG reciprocity for current algebras (2011, background): “The technical difficulties encountered in the A (2) 2n case indicate also that there must be a more general and coherent theory of local and global Weyl modules extending beyond the usual map algebras or equivariant map algebras studied, for example, in [5, 14, 28].”