LU-Decomposition of a Noncommutative Linear System and Jacobi Polynomials

Alfredo O. Brega¹ ; Leandro R. Cagliero¹

¹CIEM-FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina

Journal of Lie Theory, Tome 19 (2009) no. 3, pp. 463-481

Résumé

\def\a{{\frak a}} \def\g{{\frak g}} \def\k{{\frak k}} We obtain the LU-decomposition of a non commutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\g)^{K}\to U(\k)^{M} \otimes U(\a)$. Although this system can not be expressed as a single matrix equation with coefficients in $U(\k)$, it turns out that obtaining a triangular system equivalent to it, can be reduced to obtaining the LU-decomposition of a matrix $\widetilde M_0$ with entries in a polynomial algebra. We prove that both the L-part and U-part of $\widetilde M_0$ are expressed in terms of Jacobi polynomials. Moreover, each entry of the L-part of $\widetilde M_0$ and of its inverse is given by a single ultraspherical Jacobi polynomial. This fact yields a biorthogonality relation between the ultraspherical Jacobi polynomials.

Détail

Classification : 33C45, 22E46, 33C05, 16S30
Mots-clés : Noncommutative LU-factorization, Jacobi polynomials, K-invariants in the enveloping algebra of g, Lepowsky homomorphism