LU-decomposition of a noncommutative linear system and Jacobi polynomials

TL;DR

This paper derives an LU-decomposition for a noncommutative linear system linked to Lie algebra representations, expressing matrix entries via Jacobi polynomials and establishing a biorthogonality relation.

Contribution

It introduces a novel LU-decomposition for a noncommutative system using Jacobi polynomials, connecting matrix identities with orthogonality properties.

Findings

LU-decomposition expressed with Jacobi polynomials

Matrix identities lead to biorthogonality relations

Entries of matrices involve ultraspherical Jacobi polynomials

Abstract

In this paper we obtain the LU-decomposition of a noncommutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\lieg)^{K}\to U(\liek)^{M}\otimes U(\liea)$. This LU-decomposition can be transformed into very simple matrix identities, where the entries of the matrices involved belong to a special class of Jacobi polynomials. In particular, each entry of the L part of the original system is expressed in terms of a single ultraspherical Jacobi polynomial. In turns, these matrix identities yield a biorthogonality relation between the ultraspherical Jacobi polynomials.