Quantum $SL^+(N,\mathbb{R})$ as a locally compact quantum group

TL;DR

This paper constructs the first examples of higher-rank, purely continuous, q-deformed Lie type locally compact quantum groups using quantum cluster theory, expanding the understanding of quantum group structures.

Contribution

It introduces the first higher-rank, purely continuous, q-deformed Lie type locally compact quantum groups derived from Drinfeld-Jimbo quantization.

Findings

Construction of higher-rank quantum groups from quantum cluster theory

Application of Fock-Goncharov techniques

New examples of quantum groups in the real split Lie group setting

Abstract

We construct the first examples of purely continuous, $q$-deformed Lie type locally compact quantum groups in higher rank. They arise from Drinfeld-Jimbo quantization, at unimodular deformation parameter, of the totally positive part of higher rank split real Lie groups in type $A$. Our techniques are based on quantum cluster theory, in particular as developed through the work of Fock and Goncharov.