On the relation between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory

TL;DR

This paper establishes a direct connection between the modular double of U_q(sl(2,R)) and quantum Teichmueller theory, providing explicit representations and simplifying the understanding of fusion, braiding, and Clebsch-Gordan decompositions.

Contribution

It reveals a direct relation between the modular double of U_q(sl(2,R)) and quantum Teichmueller theory, with explicit representations and simplified derivations.

Findings

Explicit representations for fusion and braiding in quantum Teichmueller theory.

Simplified derivation of Clebsch-Gordan decomposition for the modular double.

Establishment of a direct relation between algebraic and geometric quantum structures.

Abstract

We exhibit direct relations between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory. Explicit representations for the fusion- and braiding operations of the quantum Teichmueller theory are immediate consequences. Our results include a simplified derivation of the Clebsch-Gordan decomposition for the principal series of representation of the modular double of U_q(sl(2,R)).