Quantum symmetries on noncommutative complex spheres with partial commutation relations

TL;DR

This paper introduces noncommutative complex spheres with partial commutation relations, computes their quantum symmetry groups, and explores geometric aspects of associated quantum orthogonal groups, revealing new quantum symmetries in noncommutative geometry.

Contribution

It defines noncommutative spheres with partial commutation, computes their quantum symmetry groups, and links these to quantum orthogonal groups with mixed independence.

Findings

New quantum unitary groups with partial commutation relations

Quantum symmetry groups for noncommutative spheres

Connections between quantum groups and partial commutation relations

Abstract

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.