The $C^*$-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 $C(L^{2n+1}\_q(N; m\_0,\ldots, m\_n))$ 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 $C(\mathbb W\mathbb P\_q^n(m\_0,\ldots, m\_n))$, interpreted as fixed points under the circle action on $C(S\_q^{2n+1})$, are computed under a mild assumption on the weights.