are coprime, this implies that $k_i=m\ell_{j:i}$ and $k_j=-m\ell_{i:j}$ for some $m\in{\mathbb{Z}}$. From , it follows that $z^{{\underline}{k}}=z_i^{\smash[b]{m\ell_{j:i}}}z_{j}^{\smash[b]{-m\ell_{i:j}}}$ is, modulo a multiplicative coefficient, either $(\xi_{i,j})^{|m|}$ or $(\xi_{j,i})^{|m|}$ (depending on the sign of $m$).

Clearly, $\xi_{i,i}=z_i^* z_i$ for all $i=0,\ldots,n$. We task ourself to select the class of quantum weighted projective spaces for which the elements generate the whole of ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$.

If $n=1$, and for every ${\underline}{\ell}$, it is clearly true: in this case $z^{{\underline}{k}}$ has at most length $2$, and there are no invariant monomials of length $1$. For arbitrary $n$, if $\ell_0=\ldots=\ell_n=1$ again gives a set of generators, the matrix elements of the defining projection of ${\mathbb{C}\mathrm{P}}^n_q$, as shown in [@DL10].

On the other hand, it is not difficult to find examples that do not satisfy this property. If ${\underline}{\ell}=(1,2,3)$ the element $z_0z_1z_2^*$ is irreducible in ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$ (it is not the product of invariant monomials of smaller length), hence one needs both monomials of length $2$ and $3$ to generate the algebra. Similarly, if ${\underline}{\ell}=(1,2,3,7)$ the element $z_0^2z_1z_2z_3^*$ is irreducible in ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$, and one needs elements of length $2$, $3$ and $4$ to generate the algebra.

It turns out (cf. Theorem \[thm:11\] below) that the set of elements $\xi_{i,j}$ as 