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Subalgebras of $\gc_N$ and Jacobi Polynomials

Published:2002-12-01
Printed: Dec 2002
• Alberto De Sole
• Victor G. Kac
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Abstract

We classify the subalgebras of the general Lie conformal algebra $\gc_N$ that act irreducibly on $\mathbb{C} [\partial]^N$ and that are normalized by the sl$_2$-part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials $P_n^{(-\sigma,\sigma)}$, $\sigma \in \mathbb{C}$. The connection goes both ways---we use in our classification some classical properties of Jacobi polynomials, and we derive from the theory of conformal algebras some apparently new properties of Jacobi polynomials.
 MSC Classifications: 17B65 - Infinite-dimensional Lie (super)algebras [See also 22E65] 17B68 - Virasoro and related algebras 17B69 - Vertex operators; vertex operator algebras and related structures 33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

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