${\mathbb{P}}_q({\underline}{\ell})$ in [@BF12] (with a slightly different notation for quantum spheres that the one used there). In particular, ${\mathbb{P}}_q(\,\underbrace{\!1,\ldots,1\!}_{n}\,)={\mathbb{C}\mathrm{P}}^n_q$ is a quantum projective space ‘tout court’.

Next, let $p$ be a positive integer and ${\mathbb{Z}}_p\subset {\mathrm{U}}(1)$ the subgroup of $p$-th roots of unity. We call *quantum lens space* $L_q(p;{\underline}{\ell})$ the virtual space underlying the algebra $$\label{eq:Lq} 
{\mathcal{O}}(L_q(p;{\underline}{\ell})):=\big\{a\in{\mathcal{O}}(S^{2n+1}_q):\alpha_t(a)=a \;\forall\; t\in {\mathbb{Z}}_p\big\} \;.$$

The above definition of quantum lens space in arbitrary dimension was introduced in [@HL03], in the framework of graph $C^*$-algebras.

Clearly $L_q(1,{\underline}{\ell})=S^{2n+1}_q$ and is a $*$-subalgebra of , for any $p$: $${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell})) \hookrightarrow {\mathcal{O}}(L_q(p;{\underline}{\ell})) \;.$$ We shall have a closer look at this inclusion later on in §\[sec:sei\].

As for $q=1$, we do not lose generality by working with coprime weights. Indeed,

\[lemma:obs\] For all $m\geq 1$, it holds that ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))\simeq {\mathcal{O}}({\mathbb{P}}_q(m{\underline}{\ell}))$.

Let $\alpha_{{\underline}{\ell}}(t)$ be the action in . The inclusion ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))\subset{\mathcal{O}}({\mathbb{P}}_q(m{\underline}{\ell}))$ is obvious: invariance under $\alpha_{{\underline}{\ell}}(t)$ for $t\in {\mathrm{U}}(1)$ implies invariance under $\alpha_{{\underline}{\ell}}(t^m)=\alpha_{m{\underline}{\ell}}(t)$ for $t\in {\mathrm{U}}(1)$.

The opposite inclusion follows from surjectivity of the map ${\mathrm{U}}(1)\to {\mathrm{U}}(1)$, $t\mapsto t'=t^m$: if $\alpha_{m{\underline}{\ell}}(t)(a)=a$ for $t\in {\mathrm{U}}(1)$, then $\alpha_{{\underline}{\ell}}(t')(a)=a$ for $t'=t^m\in {\mathrm{U}}(1)$.

Generators of the algebra {#sec:quattro}
-------------------------

Let us start again from the generators $\{z_i,z_i^*\}_{i=0,\ldots,n}$ of the sphere algebra ${\mathcal{O}}(S^{2n+1}_q)$. 