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

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 $U_q(s{l}_2)$ 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 $U_{q\tilde{q}}(g_R)$

References in corpus (4)

• Positive Representations of Split Real Simply-laced Quantum Groups
• On the relation between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory
• On tensor products of positive representations of split real quantum Borel subalgebra $U_{q\tilde{q}}(b_R)$
• Positive Casimir and Central Characters of Split Real Quantum Groups

Cited by in corpus (2)

• Positive Representations of Split Real Simply-laced Quantum Groups
• Continuous tensor categories from quantum groups I: algebraic aspects