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Subalgebras of $\gc_N$ and Jacobi Polynomials |
Canad. Math. Bull. 45(2002), 567-605
Published:2002-12-01
Printed: Dec 2002
Printed: Dec 2002
- Alberto De Sole
- Victor G. Kac
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] |