Quantum isometries, noncommutative spheres, and related integrals

TL;DR

This paper explores noncommutative analogues of classical spheres, detailing their construction, properties, and quantum isometry groups, advancing understanding of noncommutative geometry and quantum symmetries.

Contribution

It introduces and analyzes the half-liberated and free noncommutative spheres, detailing their structure and associated quantum isometry groups, expanding the framework of noncommutative geometry.

Findings

Construction of noncommutative spheres $S^{N-1}_{ ext{R,*}}$ and $S^{N-1}_{ ext{R,+}}$

Characterization of their main properties

Description of their quantum isometry groups

Abstract

The sphere $S^{N-1}_\mathbb R$ has a half-liberated analogue $S^{N-1}_{\mathbb R,*}$, and a free analogue $S^{N-1}_{\mathbb R,+}$. This is a presentation of the construction and main properties of these noncommutative spheres, $S^{N-1}_\mathbb R\subset S^{N-1}_{\mathbb R,*}\subset S^{N-1}_{\mathbb R,+}$, and of their quantum isometry groups.