On tensor product decomposition of positive representations of $\mathcal{U}_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$

TL;DR

This paper investigates the tensor product decomposition of positive representations of the split real quantum group U_{q ilde{q}}(sl(2,R)), deriving explicit Hilbert space decompositions using quantum dilogarithm transformations and connecting finite and infinite dimensional representation theories.

Contribution

It introduces a method to derive Hilbert space decompositions of positive representations from finite dimensional Clebsch-Gordan coefficients, extending to higher rank quantum groups.

Findings

Explicit Hilbert space decomposition via quantum dilogarithm transformations

Connection between finite and infinite dimensional representation theory

Proposed strategy for higher rank split real quantum groups

Abstract

We study the tensor product decomposition of the split real quantum group $U_{q\tilde{q}}(sl(2,R))$ from the perspective of finite dimensional representation theory of compact quantum groups. It is known that the class of positive representations of $U_{q\tilde{q}}(sl(2,R))$ is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch-Gordan coefficients of the tensor product decomposition of finite dimensional representations of the compact quantum group $U_q(sl_2)$ by solving certain functional equations and using normalization arising from tensor products of canonical basis. We propose a general strategy to deal with the tensor product decomposition for the higher rank split real quantum group $U_{q\tilde{q}}(g_R)$