LU-Decomposition of a Noncommutative Linear Systemand Jacobi Polynomials

摘要

In this paper we obtain the LU-decomposition of a non commuta-tive linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism U(g)~K→U(t)~M◎U(a) . Although this system can not be expressed as a single matrix equation with coefficients in U(t), it turns out that obtaining a triangular system equivalent to it, can be reduced to obtaining the LU-decomposition of a matrix M_0 with entries in a polynomial algebra. We prove that both the L-part and U-part of M_0 are expressed in terms of Jacobi polynomials. Moreover, each entry of the L-part of 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.