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Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$

Quentin Labriet, Loïc Poulain d'Andecy

TL;DR

This work extends the classical $sl_2$ intertwining framework to the quantum group $U_q(sl_2)$, decomposing $V_{\lambda}\otimes V_{\lambda'}$ into $\bigoplus_{n\ge 0} V_{\lambda+\lambda'+2n}$ and proving that the holographic intertwiners are governed by the little $q$-Jacobi polynomials while dual symmetry breaking intertwiners generalize Rankin--Cohen brackets to the $q$-setting. A key methodological innovation is realizing Verma modules on the quantum plane $\mathbb{C}_q[x,y]$ with $q$-commuting variables, enabling a separation of variables to a one-variable $q$-difference operator $\Theta_X$ acting on $X$ and a multiplicative $t$-part, giving explicit lowest-weight vectors and the desired decompositions. The paper builds explicit connections between two-parameter families of polynomials in two variables and the $q$-Hahn polynomials that encode Clebsch–Gordan coefficients via the $q$-Hahn algebra, and it defines $q$-Rankin–Cohen brackets as adjoints of holographic intertwiners, thereby formulating a $q$-analogue of the classical F-method. These results open a path towards higher-fold tensor products and links to Askey–Wilson structures, while providing concrete, computable formulas for $q$-intertwiners and their duals. The framework has potential implications for representation theory of quantum groups and for $q$-deformations of modular-forms-type operators in noncommutative settings.

Abstract

This paper presents explicit formulas for intertwining operators of the quantum group $U_q(sl_2)$ acting on tensor products of Verma modules. We express a first set of intertwining operators (the holographic operators) in terms of the little $q$-Jacobi polynomials, and we obtain for the dual set (the symmetry breaking operators) a $q$-deformation of the Rankin--Cohen operators. The Verma modules are realised on polynomial spaces and, interestingly, we find along the way the need to work with non-commuting variables. Explicit connections are given with the Clebsch--Gordan coefficients of $U_q(sl_2)$ expressed with the $q$-Hahn polynomials.

Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$

TL;DR

This work extends the classical intertwining framework to the quantum group , decomposing into and proving that the holographic intertwiners are governed by the little -Jacobi polynomials while dual symmetry breaking intertwiners generalize Rankin--Cohen brackets to the -setting. A key methodological innovation is realizing Verma modules on the quantum plane with -commuting variables, enabling a separation of variables to a one-variable -difference operator acting on and a multiplicative -part, giving explicit lowest-weight vectors and the desired decompositions. The paper builds explicit connections between two-parameter families of polynomials in two variables and the -Hahn polynomials that encode Clebsch–Gordan coefficients via the -Hahn algebra, and it defines -Rankin–Cohen brackets as adjoints of holographic intertwiners, thereby formulating a -analogue of the classical F-method. These results open a path towards higher-fold tensor products and links to Askey–Wilson structures, while providing concrete, computable formulas for -intertwiners and their duals. The framework has potential implications for representation theory of quantum groups and for -deformations of modular-forms-type operators in noncommutative settings.

Abstract

This paper presents explicit formulas for intertwining operators of the quantum group acting on tensor products of Verma modules. We express a first set of intertwining operators (the holographic operators) in terms of the little -Jacobi polynomials, and we obtain for the dual set (the symmetry breaking operators) a -deformation of the Rankin--Cohen operators. The Verma modules are realised on polynomial spaces and, interestingly, we find along the way the need to work with non-commuting variables. Explicit connections are given with the Clebsch--Gordan coefficients of expressed with the -Hahn polynomials.

Paper Structure

This paper contains 26 sections, 11 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Organisation of the paper.
  3. Acknowledgements
  4. Notations
  5. Tensor products of $U_q(sl_2)$-representations
  6. The Hopf algebra $U_q(sl_2)$ and its Verma modules
  7. Verma modules.
  8. Tensor product of two Verma modules
  9. The quantum plane $\mathbb{C}_q[x,y]$.
  10. Little $q$-Jacobi polynomials and decomposition of tensor products
  11. Little $q$-Jacobi polynomials
  12. Little $q$-Jacobi polynomials and lowest weight vectors
  13. The linear isomorphism $\phi$.
  14. Action of the generators in the new variables
  15. Little $q$-Jacobi polynomials and Clebsch--Gordan coefficients
  16. ...and 11 more sections

Key Result

Proposition 2.2

For any $n\geq 0$, the element: is a non-zero lowest weight vector of weight $\lambda+\lambda'+2n$, i.e. $\Delta(F) P_n^{\lambda,\lambda'}(x,y)=0$ and $\Delta(K) P_n^{\lambda,\lambda'}(x,y)=q^{\lambda+\lambda'+2n}P_n^{\lambda,\lambda'}(x,y)$.

Theorems & Definitions (28)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['propdeltaEanddecomposition']}
  • Remark 2.4
  • Corollary 2.5
  • Remark 3.1
  • Proposition 3.2
  • proof
  • ...and 18 more