Liberations and twists of real and complex spheres

TL;DR

This paper explores ten noncommutative spheres derived from real and complex spheres through liberation and twisting, characterizes them axiomatically, and computes their quantum isometry groups, proposing an extended framework with 18 spheres.

Contribution

It classifies these ten noncommutative spheres axiomatically and computes their quantum isometry groups, introducing an extended formalism with 18 spheres.

Findings

The ten noncommutative spheres are uniquely characterized under strong axioms.

Quantum isometry groups of these spheres are explicitly computed.

An extended formalism with 18 spheres is proposed.

Abstract

We study the 10 noncommutative spheres obtained by liberating, twisting, and liberating+twisting the real and complex spheres $S^{N-1}_\mathbb R,S^{N-1}_\mathbb C$. At the axiomatic level, we show that, under very strong axioms, these 10 spheres are the only ones. Our main results concern the computation of the quantum isometry groups of these 10 spheres, taken in an affine real/complex sense. We formulate as well a proposal for an extended formalism, comprising 18 spheres.