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

TL;DR

This paper investigates the topological invariance of certain quantum homogeneous spaces of types B and D, demonstrating their $q$-independence and computing their $K$-groups through algebraic and topological methods.

Contribution

It introduces a novel approach to prove $q$-invariance of specific quantum homogeneous spaces using $C^*$-algebra techniques and $K$-theory computations.

Findings

$q$-independence of $SO_q(3)$, $SO_q(5)/SO_q(3)$, $SO_q(4)/SO_q(2)$, and $SO_q(6)/SO_q(4)$.

Explicit $K$-group calculations for the studied spaces.

Generation of $C^*$-algebras by entries of fundamental matrices.

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 $SO_q(3)$, $SO_q(5)/SO_q(3)$, $SO_q(4)/SO_q(2)$, and $SO_q(6)/SO_q(4)$.