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
DOI:
https://doi.org/10.1090/S0002-9947-2011-05420-6
Published electronically:
December 29, 2011
MathSciNet review:
2888222
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Suppose a finite group acts on a scheme and a finite-dimensional Lie algebra
. The corresponding equivariant map algebra is the Lie algebra
of equivariant regular maps from
to
. 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
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:
https://doi.org/10.1090/S0002-9947-2011-05420-6
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