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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Irreducible finite-dimensional representations of equivariant map algebras


Authors: Erhard Neher, Alistair Savage and Prasad Senesi
Journal: Trans. Amer. Math. Soc. 364 (2012), 2619-2646
MSC (2010): Primary 17B10, 17B20, 17B65
Published electronically: December 29, 2011
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Abstract: Suppose a finite group acts on a scheme $ X$ and a finite-dimensional Lie algebra $ \mathfrak{g}$. The corresponding equivariant map algebra is the Lie algebra $ \mathfrak{M}$ of equivariant regular maps from $ X$ to $ \mathfrak{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 $ \mathfrak{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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Additional Information

Erhard Neher
Affiliation: Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada
Email: erhard.neher@uottawa.ca

Alistair Savage
Affiliation: Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada
Email: alistair.savage@uottawa.ca

Prasad Senesi
Affiliation: Department of Mathematics, The Catholic University of America, Washington, DC 20016
Email: senesi@cua.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05420-6
PII: S 0002-9947(2011)05420-6
Received by editor(s): April 12, 2010
Received by editor(s) in revised form: July 12, 2010
Published electronically: December 29, 2011
Additional Notes: The first and second authors gratefully acknowledge support from the NSERC through their respective Discovery grants.
The third author was partially supported by the Discovery grants of the first two authors.
Article copyright: © Copyright 2011 Erhard Neher, Alistair Savage, Prasad Senesi