In this article, we study a family of quantum homogeneous spaces, namely,<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$SO_q\left( \mathcal {N}\right) /SO_q\left( \mathcal {N}-2\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mfenced close=")" open="("> <mi mathvariant="script">N</mi> </mfenced> <mo stretchy="false">/</mo> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mfenced close=")" open="("> <mi mathvariant="script">N</mi> <mo>-</mo> <mn>2</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. By applying a two-step Zhelobenko branching rule, we show that the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C\left( SO_q\left( \mathcal {N}\right) /SO_q\left( \mathcal {N}-2\right) \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mfenced close=")" open="("> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mfenced close=")" open="("> <mi mathvariant="script">N</mi> </mfenced> <mo stretchy="false">/</mo> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mfenced close=")" open="("> <mi mathvariant="script">N</mi> <mo>-</mo> <mn>2</mn> </mfenced> </mfenced> </mrow> </math></EquationSource> </InlineEquation> are generated by the entries of the first and the last rows of the fundamental matrix of the quantum groups <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$SO_q\left( \mathcal {N}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mfenced close=")" open="("> <mi mathvariant="script">N</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. We then construct a chain of short exact sequences, and using that, we compute K-groups of these spaces with explicit generators. Invoking homogeneous <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-extension theory, we show q-independence of some intermediate <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras arising as the middle <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra of these short exact sequences. As a consequence, we get the q-invariance of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$SO_q(3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$SO_q(5)/SO_q(3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$SO_q(4)/SO_q(2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$SO_q(6)/SO_q(4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>S</mi> <msub> <mi>O</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

On Quotient Spaces of Quantized Special Orthogonal Groups and their Topological Invariance

期刊论文
发表日期：

2026年7月7日

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Akshay Bhuva,
Surajit Biswas,
Bipul Saurabh

摘要

In this article, we study a family of quantum homogeneous spaces, namely, $SO_q\left( \mathcal {N}\right) /SO_q\left( \mathcal {N}-2\right) $ $S O_{q} (N) / S O_{q} (N - 2)$ . By applying a two-step Zhelobenko branching rule, we show that the $C^*$ $C^{*}$ -algebras $C\left( SO_q\left( \mathcal {N}\right) /SO_q\left( \mathcal {N}-2\right) \right) $ $C (S O_{q} (N) / S O_{q} (N - 2))$ are generated by the entries of the first and the last rows of the fundamental matrix of the quantum groups $SO_q\left( \mathcal {N}\right) $ $S O_{q} (N)$ . We then construct a chain of short exact sequences, and using that, we compute K-groups of these spaces with explicit generators. Invoking homogeneous $C^*$ $C^{*}$ -extension theory, we show q-independence of some intermediate $C^*$ $C^{*}$ -algebras arising as the middle $C^*$ $C^{*}$ -algebra of these short exact sequences. As a consequence, we get the q-invariance of $SO_q(3)$ $S O_{q} (3)$ , $SO_q(5)/SO_q(3)$ $S O_{q} (5) / S O_{q} (3)$ , $SO_q(4)/SO_q(2)$ $S O_{q} (4) / S O_{q} (2)$ , and $SO_q(6)/SO_q(4)$ $S O_{q} (6) / S O_{q} (4)$ .