The $C^{\ast }$-algebras of quantum lens and weighted projective spaces

Abstract

It is shown that the algebra of continuous functions on the quantum $2n+1$-dimensional lens space $C(L_q^{2n+1}(N;m_0,\dots ,m_n))$ is a graph $C^{\ast }$-algebra, for arbitrary positive weights $m_0,\dots ,m_n$. 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 $C(\mathbb{W}{\mathbb{P}}_q^n(m_0,\dots ,m_n))$, interpreted as fixed points under the circle action on $C(S_q^{2n+1})$, are computed under a mild assumption on the weights.