From multiplicative unitaries to quantum groups II

TL;DR

This paper demonstrates that the core features of a C*-algebraic quantum group depend solely on its algebraic structure, not on the specific multiplicative unitary, and explores the universal dual construction without Haar weights.

Contribution

It shows that quantum group properties are determined by the algebraic pair and develops a universal dual construction independent of Haar weights.

Findings

Quantum group features depend only on (A, Δ)

Representation theory elucidates quantum group structure

Universal dual construction without Haar weights

Abstract

It is shown that all important features of a $\mathrm{C}^*$-algebraic quantum group $(A,\Delta)$ defined by a modular multiplicative $W$ depend only on the pair $(A,\Delta)$ rather than the multiplicative unitary operator $W$. The proof is based on thorough study of representations of quantum groups. As an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary - without assuming existence of Haar weights.