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arXiv · 2610.10377

Crystallization of the quantum Stiefel manifolds $SO_q(2n+1)/SO_q(2n-1)$

Surajit Biswas·Vinay Deshpande·Bipul Saurabh
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Abstract

In this paper, we introduce the crystallization $C(SO_0(2n+1)/SO_0(2n-1))$ of the $C^*$-algebra $C(SO_q(2n+1)/SO_q(2n-1))$. We show that every irreducible representation of $C(SO_0(2n+1)/SO_0(2n-1))$ arises as the limit, as $q \to 0^+$, of a family of irreducible representations of $C(SO_q(2n+1)/SO_q(2n-1))$. As a consequence, we prove that $C(SO_0(2n+1)/SO_0(2n-1))$ is the universal $C^*$-algebra generated by a finite set of generators and relations. We then construct a chain of unital essential short exact sequences of $C^*$-algebras, enabling us to compute the $K$-groups of $C(SO_0(2n+1)/SO_0(2n-1))$ and determine explicit generators for them. Using the theory of homogeneous $C^*$-extensions, we prove that $C(SO_0(2n+1)/SO_0(2n-1))$ is isomorphic to $C(SO_q(2n+1)/SO_q(2n-1))$ for every $q \in (0,1)$.

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Surajit Biswas, Vinay Deshpande, Bipul Saurabh. 2026-10-07. Crystallization of the quantum Stiefel manifolds $SO_q(2n+1)/SO_q(2n-1)$. https://arxiv.org/abs/2610.10377

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