Crystallization of the quantum Stiefel manifolds $S{O}_q(2n+1)\mathrm{/}S{O}_q(2n-1)$

In this paper, we introduce the crystallization $C(S{O}_0(2n+1)\mathrm{/}S{O}_0(2n-1))$ of the $C^{\ast }$-algebra $C(S{O}_q(2n+1)\mathrm{/}S{O}_q(2n-1))$. We show that every irreducible representation of $C(S{O}_0(2n+1)\mathrm{/}S{O}_0(2n-1))$ arises as the limit, as $q\to 0^{+}$, of a family of irreducible representations of $C(S{O}_q(2n+1)\mathrm{/}S{O}_q(2n-1))$. As a consequence, we prove that $C(S{O}_0(2n+1)\mathrm{/}S{O}_0(2n-1))$ is the universal $C^{\ast }$-algebra generated by a finite set of generators and relations. We then construct a chain of unital essential short exact sequences of $C^{\ast }$-algebras, enabling us to compute the $K$-groups of $C(S{O}_0(2n+1)\mathrm{/}S{O}_0(2n-1))$ and determine explicit generators for them. Using the theory of homogeneous $C^{\ast }$-extensions, we prove that $C(S{O}_0(2n+1)\mathrm{/}S{O}_0(2n-1))$ is isomorphic to $C(S{O}_q(2n+1)\mathrm{/}S{O}_q(2n-1))$ for every $q\in (0,1)$.