On tensor products of positive representations of split real quantum Borel subalgebra $U_{q\tilde{q}}(b_R)$

Abstract

We studied the positive representations $P_{\lambda }$ of split real quantum groups $U_{q\tilde{q}}(g_R)$ restricted to the Borel subalgebra $U_{q\tilde{q}}(b_R)$. We proved that the restriction is independent of the parameter $\lambda$. Furthermore, we prove that it can be constructed from the GNS-representation of the multiplier Hopf algebra $U_{q\tilde{q}}^{C^{\ast }}(b_R)$ constructed earlier, which enables us to decompose their tensor product using the theory of the "multiplicative unitary". This will be an essential ingredient in the construction of quantum higher Teichmüller theory from the perspective of representation theory, generalizing earlier work by Frenkel-Kim.

Revised version for publication. Introduction is rewritten. Section 6.3 is removed to shorten the paper

References in corpus (3)

• 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
• Positive representations, multiplier Hopf algebra, and continuous canonical basis

Cited by in corpus (2)

• On tensor product decomposition of positive representations of ${\mathcal{U}}_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$
• Positive Casimir and Central Characters of Split Real Quantum Groups