2009/06/30 by Erhard Neher, Alistair Savage, Prasad Senesi · 1 citation
Mathematics · #math.RT #math.AG #math.RA #msc:17B10 #msc:17B20 #msc:17B65
paper · pdf · doi:10.1090/s0002-9947-2011-05420-6
published as Trans. Amer. Math. Soc., 364 (2012), no. 5, 2619-2646 · 25 pages; v2: results generalized to schemes and arbitrary finite-dimensional g; v3: change of notation, minor typos corrected, some explanations added; v4: minor typos corrected and references updated
arxiv created 2011/12/23 · arxiv updated 2012/04/11
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.