$C(SO_q(2n+1)/SO_q(2n-1))$ as iterated torsioned quantum double suspensions of $C(\mathbb{T})$

TL;DR

The paper addresses realizing the quantum homogeneous space $SO_q(2n+1)/SO_q(2n-1)$ as an iterated $m$-torsioned quantum double suspension of $C(\mathbb{T})$, culminating in the isomorphism $C(SO_q(2n+1)/SO_q(2n-1)) \cong \Sigma^{2(n-1)} \Sigma^2_2 \Sigma^{2(n-1)} C(\mathbb{T})$ and showing independence from the deformation parameter $q$. It develops the core theory by establishing an isomorphism $\operatorname{Ext}_{\mathrm{PPV}}(Y,A) \cong \operatorname{Ext}_{\mathrm{PPV}}(Y,\Sigma^2_m A)$ and leveraging $K$-theory to identify the middle $C^*$-algebras involved in the extensions. The main result identifies the homogeneous extensions corresponding to $C(SO_q(2n+1)/SO_q(2n-1))$ as iterated suspensions applied to $C(\mathbb{T})$, yielding a description of the noncommutative space in terms of $C(\mathbb{T})$ and its suspensions. This work thereby demonstrates that the topological type of the resulting $C^*$-algebras is invariant with respect to $q$, providing a robust noncommutative-geometric model for the quantum homogeneous space.

Abstract

Let $A$ be a unital $C^*$-algebra, and let $Σ^2_m A$ denote the $m$-torsioned quantum double suspension of $A$. For $q \in (0,1)$ and $n \geq 1$, we prove that the $C^*$-algebra corresponding to the quotient space $SO_q(2n+1)/SO_q(2n-1)$ is isomorphic to $Σ^{2(n-1)} \, Σ^2_2 \, Σ^{2(n-1)} C(\mathbb{T})$. It follows as a consequence that these spaces are independent of the deformation parameter $q$.

Theorems & Definitions (22)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5
• Lemma 2.6
• Theorem 2.7
• Definition 2.8
• Proposition 2.9
• Definition 3.1