space ${\mathbb{P}}_q({\underline}\ell)$ is the quotient of the odd-dimensional quantum sphere $S^{2n+1}_q$ by a weighted action of U(1). The latter acts by automorphisms on the coordinate algebra ${\mathcal{O}}(S^{2n+1}_q)$: on generators $z_i$ just by $$\alpha_t(z_i)=t^{\ell_i}z_i \;,\qquad\forall\;t\in{\mathrm{U}}(1),\,i=0,\ldots,n$$ (cf. §\[sec:due\] for details). The fixed point algebra ${\mathcal{O}}({\mathbb{P}}_q({\underline}\ell))$ is defined to be the coordinate algebra of ${\mathbb{P}}_q({\underline}\ell)$. For $\ell_0=\ldots=\ell_n=1$, one gets in this way the quantum projective space ${\mathbb{C}\mathrm{P}}^n_q$.

Quantum weighted projective lines ${\mathbb{P}}_q(\ell_0,\ell_1)$ were studied in [@BF12], with a particular attention to quantum teardrops (for which $\ell_0=1$), and with more generality in [@AKL14]. The present paper is devoted to the case of arbitrary dimension. While the most general case of an arbitrary weight vector seems intractable at the moment, we select a particularly nice class of weight vectors which allows us to push things considerably. We focus on weight vectors ${\underline}{\ell}=(\ell_0,\ldots,\ell_n)$ for which the classical ${\mathbb{P}}({\underline}\ell)$ is homeomorphic to the ordinary projective space ${\mathbb{C}\mathrm{P}}^n$. (An algebraic characterization of these weight vectors is in §\[sec:tre\].)

By a fortunate stroke of serendipity, for such a class of weight vectors the coordinate algebra ${\mathcal{O}}({\mathbb{P}}_q({\underline}\ell))$ is generated by simple elements, of length 2 (see §\[sec:quattro\]). This gives an alternative characterization of such a class of weight vectors: one 