Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

Abstract

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of (SU(2) × SU(2), diag) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey-Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey-Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials. B Pablo Román

Explicit matrix inverses for lower triangular matrices with entries involving continuous q-ultraspherical polynomials

“…Example 4.2. The problem of finding an inverse of the matrix L β in Theorem 1.1 originally arose in [1] where the finite dimensional lower triangular matrix…”

Diabetic Patients Experience Delayed Improvement in Renal Function Post CF-LVAD Implantation

“…The problem of finding an inverse of the matrix L β in Theorem 1.1 originally arose in [1] where the finite dimensional lower triangular matrix L(x) m,n = q m−n (q 2 ; q 2 ) m (q 2 ; q 2 ) 2n+1 (q 2 ; q 2 ) m+n+1 (q 2 ; q 2 ) n C m−n (x; q 2n+2 |q 2 ), with 0 ≤ n ≤ m ≤ N , for arbitrary N ∈ N appears. The problem of finding an inverse of the matrix L β in Theorem 1.1 originally arose in [1] where the finite dimensional lower triangular matrix L(x) m,n = q m−n (q 2 ; q 2 ) m (q 2 ; q 2 ) 2n+1 (q 2 ; q 2 ) m+n+1 (q 2 ; q 2 ) n C m−n (x; q 2n+2 |q 2 ), with 0 ≤ n ≤ m ≤ N , for arbitrary N ∈ N appears.…”

Branching Rules for Finite-Dimensional U q ( 𝖘 𝖚 ( 3 ) ) $\mathcal {U}_{q}(\mathfrak {su}(3))$ -Representations with Respect to a Right Coideal Subalgebra

“…One recent extension of this situation [1], where higher-dimensional representations of the coideal subalgebra B are involved, arises with the study of matrix-valued spherical functions of the quantum analogue of (SU(2) × SU(2), SU (2)) where the subgroup is diagonally embedded. The quantum symmetric pair is given by the quantised universal enveloping algebra of U q (g), where g = su(2) ⊕ su (2), and a right coideal subalgebra B that can be identified with U q (su (2)).…”

“…One of the first technical difficulties that one runs into in order to extend the results of [1] to more general quantum symmetric pairs is the lack of the explicit branching rules for finite-dimensional U q (g)-representations with respect to a right coideal subalgebra. In this paper we deal with this problem for the quantised universal enveloping algebra U q (su (3)) with a right coideal subalgebra B as in Kolb [13].…”