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

 

Multiloop realization of extended affine Lie algebras and Lie tori


Authors: Bruce Allison, Stephen Berman, John Faulkner and Arturo Pianzola
Journal: Trans. Amer. Math. Soc. 361 (2009), 4807-4842
MSC (2000): Primary 17B65; Secondary 17B67, 17B70
Published electronically: April 21, 2009
MathSciNet review: 2506428
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Abstract: An important theorem in the theory of infinite dimensional Lie algebras states that any affine Kac-Moody algebra can be realized (that is to say constructed explicitly) using loop algebras. In this paper, we consider the corresponding problem for a class of Lie algebras called extended affine Lie algebras (EALAs) that generalize affine algebras. EALAs occur in families that are constructed from centreless Lie tori, so the realization problem for EALAs reduces to the realization problem for centreless Lie tori. We show that all but one family of centreless Lie tori can be realized using multiloop algebras (in place of loop algebras). We also obtain necessary and sufficient conditions for two centreless Lie tori realized in this way to be isotopic, a relation that corresponds to isomorphism of the corresponding families of EALAs.


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Additional Information

Bruce Allison
Affiliation: Department of Mathematics and Statistics, University of Victoria, PO Box 3060 STN CSC, Victoria, British Columbia, Canada V8W 3R4
Email: ballison@uvic.ca

Stephen Berman
Affiliation: Saskatoon, Saskatchewan, Canada
Email: sberman@shaw.ca

John Faulkner
Affiliation: Department of Mathematics, University of Virginia, Kerchof Hall, P.O. Box 400137, Charlottesville, Virginia 22904-4137
Email: jrf@virginia.edu

Arturo Pianzola
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: a.pianzola@ualberta.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04828-4
PII: S 0002-9947(09)04828-4
Received by editor(s): September 7, 2007
Published electronically: April 21, 2009
Additional Notes: The first and fourth authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.