Notes on quantum weighted projective spaces and multidimensional teardrops

Abstract

It is shown that the coordinate algebra of the quantum $2n+1$-dimensional lens space $\mathcal{O}(L_q^{2n+1}({\prod }_{i=0}^n{m}_i;m_0,\dots ,m_n))$ is a principal $\mathbb{Z}$-comodule algebra or the coordinate algebra of a circle principal bundle over the weighted quantum projective space ${\mathbb{WP}}_q^n(m_0,\dots ,m_n)$. Furthermore, the weighted $U(1)$-action or the $\mathbb{CZ}$-coaction on the quantum odd dimensional sphere algebra $\mathcal{O}(S_q^{2n+1})$ that defines ${\mathbb{WP}}_q^n(1,m_1,\dots ,m_n)$ is free or principal. Analogous results are proven for quantum real weighted projective spaces ${\mathbb{RP}}_q^{2n}(m_0,\dots ,m_n)$. The $K$-groups of ${\mathbb{WP}}_q^n(1,\dots ,1,m)$ and ${\mathbb{RP}}_q^{2n}(1,\dots ,1,m)$ and the $K_1$-group of $L_q^{2n+1}(N;m_0,\dots ,m_n)$ are computed

14 pages; v2 minor revisions

References in corpus (10)

• Pimsner algebras and Gysin sequences from principal circle actions
• Quantum principal bundles over quantum real projective spaces
• Quantum teardrops
• Fredholm Modules for Quantum Euclidean Spheres
• Corners of graph algebras
• Quantum weighted projective and lens spaces
• Bounded and unbounded Fredholm modules for quantum projective spaces
• On the Smoothness of the Noncommutative Pillow and Quantum Teardrops
• Circle actions on a quantum Seifert manifold
• Circle and line bundles over generalized Weyl algebras

Cited by in corpus (3)

• Pimsner algebras and circle bundles
• Topics in Noncommutative Geometry
• Gysin exact sequences for quantum weighted lens spaces