Liberations and twists of real and complex spheres

Abstract

We study the 10 noncommutative spheres obtained by liberating, twisting, and liberating+twisting the real and complex spheres $S_{\mathbb{R}}^{N-1},S_{\mathbb{C}}^{N-1}$. 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.

38 pages

References in corpus (8)

• Why the Standard Model
• Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory
• Quantum Group of Isometries in Classical and Noncommutative Geometry
• The Noncommutative Geometry of the Quantum Projective Plane
• Isomorphisms and Fusion Rules of Orthogonal Free Quantum Groups and their Complexifications
• On quantum symmetries of compact metric spaces
• Quantum rigidity of negatively curved manifolds
• Existence and examples of quantum isometry group for a class of compact metric spaces

Cited by in corpus (6)

• Unitary easy quantum groups: geometric aspects
• Matrix models for noncommutative algebraic manifolds
• Feynman-Kac perturbation of $C^{\ast }$ quantum stochastic flows
• Super-easy quantum groups with arbitrary parameters
• Quantum symmetries on noncommutative complex spheres with partial commutation relations
• Modelling questions for transitive quantum groups