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Proceedings of the American Mathematical Society

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



Four-point affine Lie algebras

Author: Murray Bremner
Journal: Proc. Amer. Math. Soc. 123 (1995), 1981-1989
MSC: Primary 17B67; Secondary 33C45, 33C50
MathSciNet review: 1249871
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Abstract: We consider Lie algebras of the form $ \mathfrak{g} \otimes R$ where $ \mathfrak{g}$ is a simple complex Lie algebra and $ R = \mathbb{C}[s,{s^{ - 1}},{(s - 1)^{ - 1}},{(s - a)^{ - 1}}]$ for $ a \in \mathbb{C} - \{ 0,1\} $. After showing that R is isomorphic to a quadratic extension of the ring $ \mathbb{C}[t,{t^{ - 1}}]$ of Laurent polynomials, we prove that $ g \otimes R$ is a quasi-graded Lie algebra with a triangular decomposition. We determine the universal central extension of $ \mathfrak{g} \otimes R$ and show that the cocycles defining it are closely related to ultraspherical (Gegenbauer) polynomials.

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Article copyright: © Copyright 1995 American Mathematical Society

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