K-groups of the quantum homogeneous space $SU_{q}(n)/SU_{q}(n-2)$

TL;DR

This paper computes the K-theory groups of certain quantum homogeneous spaces derived from quantum special unitary groups, revealing new topological invariants and properties of their fundamental unitaries.

Contribution

It provides the first explicit computation of K-groups for the spaces $SU_q(n)/SU_q(n-2)$, extending the understanding of their topological structure.

Findings

K-groups of $SU_q(n)/SU_q(n-2)$ computed for all $n \\ge 3$

Fundamental unitary in quantum $SU(3)$ is nontrivial in $K_1$

Identifies the fundamental unitary as a unimodular element in $K_1$

Abstract

Quantum Steiffel manifolds were introduced by Vainerman and Podkolzin in \cite{VP}. They classified the irreducible representations of their underlying $C^*$-algebras. Here we compute the K groups of the quantum homogeneous spaces $SU_{q}(n)/SU_{q}(n-2), n\ge 3$. Specializing to the case $n=3$ we show that the fundamental unitary for quantum $SU(3)$ is nontrivial and is a unimodular element in $K_1$.