Maximal torus theory for compact quantum groups

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 $\{\widehat{\Gamma }_{Q}|Q\in U_{N}\}$.