Quantum alpha-determinants and q-deformed hypergeometric polynomials

2009/02/26 by Kazufumi Kimoto, Kimoto, Kazufumi
Mathematics · Physics and Astronomy · #20G42 #33C20 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0902.4608

openalex publication_date 2009/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The quantum α-determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic U_q(\mathfraksl₂)-submodules of the quantum matrix algebra A_q(Mat₂) generated by the powers of the quantum α-determinant. For such a cyclic module, there exists a collection of polynomials which describe the irreducible decomposition of it in the following manner: (i) each polynomial corresponds to a certain irreducible U_q(\mathfraksl₂)-module, (ii) the cyclic module contains an irreducible submodule if the parameter is a root of the corresponding polynomial. These polynomials are given as a q-deformation of the hypergeometric polynomials. This is a quantum analogue of the result obtained in our previous work [K. Kimoto, S. Matsumoto and M. Wakayama, Alpha-determinant cyclic modules and Jacobi polynomials, to appear in Trans. Amer. Math. Soc.].

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