Symmetries of quantum spaces. Subgroups and quotient spaces of quantum $SU(2)$ and $SO(3)$ groups

TL;DR

This paper analyzes the structure of quantum symmetries by decomposing actions of compact quantum groups on quantum spaces, providing formulas for multiplicities, and describing subgroups and quotients of quantum SU(2) and SO(3).

Contribution

It introduces a decomposition method for quantum group actions and characterizes subgroups and quotients of quantum SU(2) and SO(3), advancing understanding of quantum symmetries.

Findings

Decomposition of quantum group actions into irreducible representations.

Formulas for multiplicities in quotient quantum spaces.

Descriptions of subgroups and quotient spaces of quantum SU(2) and SO(3).

Abstract

We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spaces of quantum SU(2) and SO(3) groups.