Connectedness and Gaussian Parts for Compact Quantum Groups

TL;DR

This paper introduces the Gaussian part of a compact quantum group, explores its properties, and characterizes it for classical and deformed groups, revealing connections to strong connectedness.

Contribution

It defines the Gaussian part of a compact quantum group and characterizes it for various classes, linking it to strong connectedness and expanding understanding of quantum group structure.

Findings

Gaussian part is contained in the Kac part

Characterization of Gaussian parts for classical and deformed groups

Examples of strongly connected and disconnected quantum groups

Abstract

We introduce the Gaussian part of a compact quantum group $\mathbb{G}$, namely the largest quantum subgroup of $\mathbb{G}$ supporting all the Gaussian functionals of $\mathbb{G}$. We prove that the Gaussian part is always contained in the Kac part, and characterise Gaussian parts of classical compact groups, duals of classical discrete groups and $q$-deformations of compact Lie groups. The notion turns out to be related to a new concept of "strong connectedness" and we exhibit several examples of both strongly connected and totally strongly disconnected compact quantum groups.