Beurling-Fourier algebras of compact quantum groups: characters and finite dimensional representations

TL;DR

This paper explores weighted Fourier algebras of compact quantum groups, analyzing their spectral properties and finite dimensional representations, revealing connections to classical subgroups and complexification structures.

Contribution

It introduces a spectral analysis of weighted Fourier algebras on quantum groups and demonstrates their ability to detect complexification structures, especially for $SU_q(2)$.

Findings

Spectral analysis links to classical subgroups and their complexification.

Weighted Fourier algebras can detect complexification structures.

Application to $SU_q(2)$ reveals connection to quantum Lorentz group.

Abstract

In this paper we study weighted versions of Fourier algebras of compact quantum groups. We focus on the spectral aspects of these Banach algebras in two different ways. We first investigate their Gelfand spectrum, which shows a connection to the maximal classical closed subgroup and its complexification. Secondly, we study specific finite dimensional representations coming from the complexification of the underlying quantum group. We demonstrate that the weighted Fourier algebras can detect the complexification structure in the special case of $SU_q(2)$, whose complexification is the quantum Lorentz group $SL_q(2,\mathbb{C})$.