The $C^*$-algebras of quantum lens and weighted projective spaces

TL;DR

This paper demonstrates that the algebra of continuous functions on quantum lens spaces and weighted projective spaces can be described as graph C*-algebras, enabling explicit computation of their K-groups.

Contribution

It provides a new graph algebraic description of quantum lens and weighted projective spaces, facilitating K-theory calculations and deepening understanding of their structure.

Findings

Quantum lens space algebras are graph C*-algebras.

K-groups of specific quantum lens spaces are explicitly computed.

K-groups of quantum weighted projective spaces are determined under mild conditions.

Abstract

It is shown that the algebra of continuous functions on the quantum $2n+1$-dimensional lens space $C(L^{2n+1}_q(N; m_0,\ldots, m_n))$ is a graph $C^*$-algebra, for arbitrary positive weights $ m_0,\ldots, m_n$. The form of the corresponding graph is determined from the skew product of the graph which defines the algebra of continuous functions on the quantum sphere $S_q^{2n+1}$ and the cyclic group $\mathbb{Z}_N$, with the labelling induced by the weights. Based on this description, the K-groups of specific examples are computed. Furthermore, the K-groups of the algebras of continuous functions on quantum weighted projective spaces $C(\mathbb{WP}_q^n(m_0,\ldots, m_n))$, interpreted as fixed points under the circle action on $C(S_q^{2n+1})$, are computed under a mild assumption on the weights.