Uniqueness results for noncommutative spheres and projective spaces

TL;DR

This paper establishes uniqueness results for certain noncommutative spheres, projective spaces, and quantum groups, showing they are the only structures satisfying specific combinatorial axioms within their categories.

Contribution

It extends known classification results from quantum groups to noncommutative geometric spaces and projective quantum groups under combinatorial axioms.

Findings

Classification of noncommutative spheres and projective spaces

Uniqueness of quantum groups under combinatorial axioms

Extension of known quantum group results to geometric structures

Abstract

It is known that, under strong combinatorial axioms, $O_N\subset O_N^*\subset O_N^+$ are the only orthogonal quantum groups. We prove here similar results for the noncommutative spheres $S^{N-1}_\mathbb R\subset S^{N-1}_{\mathbb R,*}\subset S^{N-1}_{\mathbb R,+}$, the noncommutative projective spaces $P^{N-1}_\mathbb R\subset P^{N-1}_\mathbb C\subset P^{N-1}_+$, and the projective orthogonal quantum groups $PO_N\subset PO_N^*\subset PO_N^+$.