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Four-point affine Lie algebras. (English) Zbl 0833.17023

Lie algebras of the form \({\mathfrak g} \otimes R\) where \({\mathfrak g}\) is a simple complex Lie algebra and \(R= \mathbb{C} [s, s^{-1}, (s- 1)^{-1}, (sa )^{-1} ]\) are considered. The algebra \(R\) can be identified with the algebra of meromorphic functions on \(\mathbb{P}^1 (\mathbb{C})\) which are holomorphic outside the points \(\infty\), 0, 1, and \(a\). It is shown that \(R\) is isomorphic to a quadratic extension of the algebra of Laurent polynomials and that \({\mathfrak g} \otimes R\) is an almost-graded algebra with a triangular decomposition. Its 3-dimensional universal central extension is determined very explicitly. Such central extensions are multi-point generalizations of affine Kac-Moody algebras. The cocycles are given. They are closely related to ultraspherical (Gegenbauer) polynomials.

MSC:

17B65 Infinite-dimensional Lie (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
14H99 Curves in algebraic geometry
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References:

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