Projective quantum spaces

TL;DR

This paper introduces quantum analogs of classical geometric spaces such as spheres, projective spaces, and Grassmann manifolds, demonstrating their structure as homogeneous quantum spaces and quantum principal bundles.

Contribution

It constructs and analyzes quantum versions of classical geometric spaces associated with $SU_q(n)$, establishing their properties as homogeneous quantum spaces and quantum principal bundles.

Findings

Defined quantum spheres $S_q^{2n-1}$, projective spaces $CP_q^{n-1}$, and Grassmann manifolds $G_k(C_q^n)$.

Proved these algebras are homogeneous quantum spaces of quantum groups.

Showed these spaces form quantum principal bundles as per Brzezinski and Majid's framework.

Abstract

Associated to the standard $SU_{q}(n)$ R-matrices, we introduce quantum spheres $S_{q}^{2n-1}$, projective quantum spaces $CP_{q}^{n-1}$, and quantum Grassmann manifolds $G_{k}(C_{q}^{n})$. These algebras are shown to be homogeneous quantum spaces of standard quantum groups and are also quantum principle bundles in the sense of T Brzezinski and S. Majid (Comm. Math. Phys. 157,591 (1993)).