Associated noncommutative vector bundles over the Vaksman-Soibelman quantum complex projective spaces

TL;DR

This paper constructs noncommutative vector bundles over Vaksman-Soibelman quantum projective spaces and proves they generate the even K-theory group, extending classical geometric concepts into the quantum setting.

Contribution

It introduces a method to generate K-theory elements via associated noncommutative vector bundles in quantum projective spaces.

Findings

Noncommutative vector bundles generate the K-theory group.

Extension of circle actions to quantum group actions.

Explicit construction of generators for K-theory.

Abstract

By a diagonal embedding of $U(1)$ in $SU_q(m)$, we prolongate the diagonal circle action on the Vaksman-Soibelman quantum sphere $S^{2n+1}_q$ to the $SU_q(m)$-action on the prolongated bundle. Then we prove that the noncommutative vector bundles associated via the fundamental representation of $SU_q(m)$, for $m\in\{2,\ldots,n\}$, yield generators of the even K-theory group of the C*-algebra of the Vaksman-Soibelman quantum complex projective space $\mathbb{C}{\rm P}^n_q$.