Journal of Lie Theory

no. 3
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, Volume 19 (2009) no. 3, pp. 463-481
\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.
DOI: 10.5802/jolt.563
Classification: 33C45, 22E46, 33C05, 16S30
Keywords: Noncommutative LU-factorization, Jacobi polynomials, K-invariants in the enveloping algebra of g, Lepowsky homomorphism

Alfredo O. Brega  1 ; Leandro R. Cagliero  1

1 CIEM-FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina 
Alfredo O. Brega; Leandro R. Cagliero. LU-Decomposition of a Noncommutative Linear System and Jacobi Polynomials. Journal of Lie Theory, Volume 19 (2009) no. 3, pp. 463-481. doi: 10.5802/jolt.563
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     author = {Alfredo O. Brega and Leandro R. Cagliero},
     title = {LU-Decomposition of a {Noncommutative} {Linear} {System} and {Jacobi} {Polynomials}},
     journal = {Journal of Lie Theory},
     pages = {463--481},
     year = {2009},
     volume = {19},
     number = {3},
     doi = {10.5802/jolt.563},
     zbl = {1179.33009},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.563/}
}
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