Haar measure on a locally compact quantum group

TL;DR

This paper analyzes the Haar weight in a specific non-compact quantum group, demonstrating its structure aligns with the modern definition of locally compact quantum groups and clarifying its relation to deformation quantization.

Contribution

It provides a detailed analysis of Haar weight for a particular quantum group, confirming its status as a locally compact quantum group per recent definitions.

Findings

Confirmed (A, Δ) as a locally compact quantum group

Clarified the relationship between original construction and general theory

Established the Haar weight structure for the quantum group

Abstract

In the general theory of locally compact quantum groups, the notion of Haar measure (Haar weight) plays the most significant role. The aim of this paper is to carry out a careful analysis regarding Haar weight, in relation to general theory, for the specific non-compact quantum group (A,\Delta) constructed earlier by the author. In this way, one can show that (A,\Delta) is indeed a ``(C*-algebraic) locally compact quantum group'' in the sense of the recently developed definition given by Kustermans and Vaes. Attention will be given to pointing out the relationship between the original construction (obtained by deformation quantization) and the structure maps suggested by general theory.