Quantum isometries of noncommutative polygonal spheres

TL;DR

This paper investigates the quantum symmetries of noncommutative polygonal spheres, extending classical geometric concepts to quantum and noncommutative settings, including complex analogues and various deformations.

Contribution

It provides a classification of quantum isometries for noncommutative polygonal spheres and their deformations, advancing understanding of symmetries in noncommutative geometry.

Findings

Quantum isometries of polygonal spheres are classified.

Results include noncommutative and deformed analogues.

Complex sphere versions are also analyzed.

Abstract

The real sphere $S^{N-1}_\mathbb R$ appears as increasing union, over $d\in\{1,...,N\}$, of its "polygonal" versions $S^{N-1,d-1}_\mathbb R=\{x\in S^{N-1}_\mathbb R|x_{i_0}... x_{i_d}=0,\forall i_0,...,i_d\ {\rm distinct}\}$. Motivated by general classification questions for the undeformed noncommutative spheres, smooth or not, we study here the quantum isometries of $S^{N-1,d-1}_\mathbb R$, and of its various noncommutative analogues, obtained via liberation and twisting. We discuss as well a complex version of these results, with $S^{N-1}_\mathbb R$ replaced by the complex sphere $S^{N-1}_\mathbb C$.