Subalgebras of $\gc_N$ and Jacobi polynomials

TL;DR

This paper classifies certain subalgebras of the Lie conformal algebra _N, linking their structure to classical Jacobi polynomials, and uncovers new properties of these polynomials through conformal algebra theory.

Contribution

It provides a classification of subalgebras of _N related to _N and reveals new properties of Jacobi polynomials derived from conformal algebra analysis.

Findings

Classification of subalgebras acting irreducibly on []^N

Establishment of a connection between subalgebras and Jacobi polynomials

Discovery of new properties of Jacobi polynomials from conformal algebra theory

Abstract

We classify the subalgebras of the general Lie conformal algebra $\gc_N$ that act irreducibly on $\C[\partial]^N$ and that are normalized by the $\operatorname{sl}_2$--part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials $P_n^{(-\sigma,\sigma)}$, $\sigma\in\C$. 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.