Uniqueness results for noncommutative spheres and projective spaces

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 $PO_{N}\subset PO_{N}^{*}\subset PO_{N}^{+}$.