only (cf. Lemma \[lemma:obs\] for the quantum case).

If ${\underline}{\ell}=(\ell_0,\ldots,\ell_n)$ is a weight vector, we denote by ${\underline}{\ell}^{\,\sharp}$ the weight vector whose $i$-th component is equal to $\prod_{j\neq i}\ell_j$, for all $i=0,\ldots,n$.

For all $p\geq 1$ and $0\leq k\leq n$, we further define an operation ${\mathcal}{M}_k(p)$ on weight vectors as follows: we let ${\mathcal}{M}_k(p){\underline}{\ell}$ be the weight vector whose $i$-th component is equal to $\ell_i$ if $i=k$, and to $p\ell_i$ otherwise.

The left inverse of ${\mathcal}{M}_k(p)$ is the operation ${\mathcal}{D}_k(p)$ partially defined as follows: if $p$ divides $\ell_j$ for all $j\neq k$, then ${\mathcal}{D}_k(p){\underline}{\ell}$ is defined as the weight vector whose $i$-th component is equal to $\ell_i$ if $i=k$, and to $p^{-1}\ell_i$ otherwise.

The map ${\underline}{\ell}\mapsto{\underline}{\ell}^{\,\sharp}$ is not exactly an involution, but it satisfies $$({\underline}{\ell}^{\,\sharp})^{\,\sharp}=m \hspace{1pt} {\underline}{\ell}$$ with $m:=(\ell_0\ell_1\cdots\ell_n)^{n-1}$. Therefore, from the above mentioned fact that ${\mathbb{P}}({\underline}{\ell})\simeq{\mathbb{P}}(m{\underline}{\ell})$, every isomorphism class of weighted projective spaces can be obtained from a weight that is in the image of this map. A similar result holds for the quantum case as well, cf. Lemma \[lemma:obs\] below.

Let ${\underline}{\ell}$ be a weight vector. The *multiplication* ${\mathcal}{M}_k(p)$ is *admissible* on ${\underline}{\ell}$ if $p$ is a prime number and is not a divisor of $\ell_k$. The *division* 