Maximal torus theory for compact quantum groups

TL;DR

This paper introduces a framework for understanding compact quantum groups through a family of subgroup analogs called maximal tori, and explores conjectures linking their algebraic and analytic properties.

Contribution

It proposes a series of conjectures connecting the properties of compact quantum groups to those of their associated maximal tori subgroups.

Findings

Introduction of a canonical family of group dual subgroups for compact quantum groups

Formulation of conjectures relating algebraic and analytic properties

Establishment of a new perspective on quantum group structure

Abstract

Associated to any compact quantum group $G\subset U_N^+$ is a canonical family of group dual subgroups $\widehat{\Gamma}_Q\subset G$, parametrized by unitaries $Q\in U_N$, playing the role of "maximal tori" for $G$. We present here a series of conjectures, relating the various algebraic and analytic properties of $G$ to those of the family $\{\widehat{\Gamma}_Q|Q\in U_N\}$.