Topological invariance of quantum homogeneous spaces of type $B$ and $D$

Abstract

In this article, we study two families of quantum homogeneous spaces, namely, $S{O}_q(2n+1)/S{O}_q(2n-1)$, and $S{O}_q(2n)/S{O}_q(2n-2)$. By applying a two-step Zhelobenko branching rule, we show that the $C^{\ast }$-algebras $C(S{O}_q(2n+1)/S{O}_q(2n-1))$, and $C(S{O}_q(2n)/S{O}_q(2n-2))$ are generated by the entries of the first and the last rows of the fundamental matrix of the quantum groups $S{O}_q(2n+1)$, and $S{O}_q(2n)$, 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 $C^{\ast }$-extension theory, we show $q$-independence of some intermediate $C^{\ast }$-algebras arising as the middle $C^{\ast }$-algebra of these short exact sequences. As a consequence, we get the $q$-invariance of $S{O}_q(3)$, $S{O}_q(5)/S{O}_q(3)$, $S{O}_q(4)/S{O}_q(2)$, and $S{O}_q(6)/S{O}_q(4)$.

Presentation revised. Theorem 3.6 and Proposition 4.6 added. Lemma 4.6 removed. Theorem 4.7 corrected accordingly