Uniqueness results for noncommutative spheres and projective spaces

Abstract

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

13 pages