Positive representations, multiplier Hopf algebra, and continuous canonical basis

TL;DR

This paper explores the use of multiplier Hopf algebras in positive representations of split real quantum groups, aiming to establish a continuous canonical basis that could advance harmonic analysis and tensor product closure.

Contribution

It introduces a new framework combining multiplier Hopf algebras with positive representations and continuous canonical bases for split real quantum groups.

Findings

Proposes a continuous version of Lusztig-Kashiwara's canonical basis.

Suggests potential proof of positive representations' closure under tensor products.

Lays groundwork for harmonic analysis in locally compact quantum groups.

Abstract

We introduce the language of multiplier Hopf algebra in the context of positive representations of split real quantum groups, and discuss its applications with a continuous version of Lusztig-Kashiwara's canonical basis, which may provide a key to prove the closure of the positive representations under tensor products, and harmonic analysis of quantized algebra of functions in the sense of locally compact quantum groups.