Topological invariance of quantum homogeneous spaces of type $B$ and $D$
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866912967981793280 |
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| author | Bhuva, Akshay Biswas, Surajit Saurabh, Bipul |
| author_facet | Bhuva, Akshay Biswas, Surajit Saurabh, Bipul |
| contents | 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 $SO_q(3)$, $SO_q(5)/SO_q(3)$, $SO_q(4)/SO_q(2)$, and $SO_q(6)/SO_q(4)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological invariance of quantum homogeneous spaces of type $B$ and $D$ Bhuva, Akshay Biswas, Surajit Saurabh, Bipul Quantum Algebra 58B34, 46L80, 19K33 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 $SO_q(3)$, $SO_q(5)/SO_q(3)$, $SO_q(4)/SO_q(2)$, and $SO_q(6)/SO_q(4)$. |
| title | Topological invariance of quantum homogeneous spaces of type $B$ and $D$ |
| topic | Quantum Algebra 58B34, 46L80, 19K33 |
| url | https://arxiv.org/abs/2406.19074 |