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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The universal central extension of the three-point $ \mathfrak{sl}_2$ loop algebra


Authors: Georgia Benkart and Paul Terwilliger
Journal: Proc. Amer. Math. Soc. 135 (2007), 1659-1668
MSC (2000): Primary 17B37
DOI: https://doi.org/10.1090/S0002-9939-07-08765-5
Published electronically: January 8, 2007
MathSciNet review: 2286073
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Abstract: We consider the three-point loop algebra,

$\displaystyle L= \mathfrak{sl}_2\otimes \mathbb{K} \lbrack t, t^{-1}, (t-1)^{-1}\rbrack,$

where $ \mathbb{K}$ denotes a field of characteristic 0 and $ t$ is an indeterminate. The universal central extension $ \widehat L$ of $ L$ was determined by Bremner. In this note, we give a presentation for $ \widehat L$ via generators and relations, which highlights a certain symmetry over the alternating group $ A_4$. To obtain our presentation of $ \widehat L$, we use the realization of $ L$ as the tetrahedron Lie algebra.


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Additional Information

Georgia Benkart
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: benkart@math.wisc.edu

Paul Terwilliger
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: terwilli@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08765-5
Keywords: ${\mathfrak{sl}}_2$ loop algebra, universal central extension, tetrahedron Lie algebra, Onsager Lie algebra.
Received by editor(s): December 17, 2005
Received by editor(s) in revised form: February 24, 2006
Published electronically: January 8, 2007
Additional Notes: The first author’s support from NSF grant #DMS–0245082 is gratefully acknowledged.
Communicated by: Dan M. Barbasch
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.