which is a deformation of the algebraic equations defining the lens space $L(p;{\underline}{\ell})$.

For $n=1$, the algebra ${\mathcal{O}}(L_q(p;1,p))$ is isomorphic to the abstract unital $*$-algebra generated by elements as in with relations as in Proposition \[prop:relLens\] (essentially meaning there are no additional relations among the generators). While it ought to be possible to establish an analogous statement for general $n$ and any weight ${\underline}{\ell}$ as in Theorem \[thm:13\], such a result is not needed in the following.

Irreducible representations {#sec:irreps}
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Irreducible representation of quantum spheres were constructed in [@HL04]. From these, by restriction one gets irreducible representations of quantum lens and weighted projective spaces. They will be used in the next section to construct Fredholm modules.

Denote by ${\left|\smash[t]{\vec{k}}\right>}$ the canonical orthonormal basis of $\ell^2({\mathbb{N}}^n)$, where $\vec{k}=(k_1,\ldots,k_n)\in{\mathbb{N}}^n$, and by $\vec{e}_i$ the vector with $i$-th component equal to $1$ and all the others equal to zero (for $i=1,\ldots,n$). A faithful representation of ${\mathcal{O}}(S^{2n+1}_q)$ on $\ell^2({\mathbb{N}}^n)$ is given on generators by $$\begin{aligned}
z_i{\left|\smash[t]{\vec{k}}\right>} &=q^{k_1+\ldots+k_i}\sqrt{1-q^{2(k_{i+1}+1)}}{\left|\smash[t]{\vec{k}+\vec{e}_{i+1}}\right>} \;, \qquad \textup{for} \;\; 0\leq i<n \, , \\
z_n{\left|\smash[t]{\vec{k}}\right>} &=q^{k_1+\ldots+k_n}{\left|\smash[t]{\vec{k}}\right>} \, ,\end{aligned}$$ where we omit the representation symbol. This is the representation $\psi_1^{(2n+1)}$ of [@HL04], modulo a renaming of the generators and a redefinition of the 