Topological invariance of quantum homogeneous spaces of type $B$ and $D$
Abstract
In this article, we study two families of quantum homogeneous spaces, namely, $SO_q(2n+1)/SO_q(2n-1)$, and $SO_q(2n)/SO_q(2n-2)$. By applying a two-step Zhelobenko branching rule, we show that the $C^*$-algebras $C(SO_q(2n+1)/SO_q(2n-1))$, and $C(SO_q(2n)/SO_q(2n-2))$ are generated by the entries of the first and the last rows of the fundamental matrix of the quantum groups $SO_q(2n+1)$, and $SO_q(2n)$, respectively. We then construct a chain of short exact sequences, and using that, we compute $K$-groups of these spaces with explicit generators. Invoking homogeneous $C^*$-extension theory, we show $q$-independence of some intermediate $C^*$-algebras arising as the middle $C^*$-algebra of these short exact sequences. As a consequence, we get the $q$-invariance of $SO_q(5)/SO_q(3)$ and $SO_q(6)/SO_q(4)$.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.19074
- arXiv:
- arXiv:2406.19074
- Bibcode:
- 2024arXiv240619074B
- Keywords:
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- Mathematics - Quantum Algebra;
- 58B34;
- 46L80;
- 19K33