To simplify the notations, we also denote by $z_i^{-1}$ the adjoint $z_i^*$ (there is no ambiguity, since $z_i$ is not invertible). For ${\underline}{k}=(k_0,\ldots,k_n)\in{\mathbb{Z}}^{n+1}$, let $$z^{{\underline}{k}}:=z_0^{k_0}z_1^{k_1}\ldots z_n^{k_n} \;.$$ Let ${\underline}{x}\cdot{\underline}{y}=x_0y_0+\ldots+x_ny_n$ be the Euclidean inner product. Next lemma is true for an arbitrary weight vector ${\underline}{\ell}$.

\[lemma:2\] For any weight vector ${\underline}{\ell}$, a set of generators for the weighted projective space algebra ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$ is given by the elements: $$\begin{aligned}
z_i^*z_i \;,\quad &\forall\;i=0,\ldots,n \\[2pt]
z^{{\underline}{k}} \;,\quad &\forall\;{\underline}{k}\in{\mathbb{Z}}^{n+1}\; \quad \textup{s.t.}\quad \;{\underline}{\ell}\cdot{\underline}{k}=0 \,  .\end{aligned}$$

A linear basis of ${\mathcal{O}}(S^{2n+1}_q)$ is given by monomials $$z_0^{j_0}z_1^{j_1}(z_1^*)^{k_1}\ldots z_n^{j_n}(z_n^*)^{k_n} \;,\qquad
(z_0^*)^{k_0}z_1^{j_1}(z_1^*)^{k_1}\ldots z_n^{j_n}(z_n^*)^{k_n} \;,$$ where $j_0,\ldots,j_n,k_0,\ldots,k_n$ are non-negative integers. Using the commutation rules of $S^{2n+1}_q$, these can be always rewritten as sums of products of elements $z_i^*z_i$, which clearly belong to ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$, and elements $z^{{\underline}{k}}$. Under the action : $$z^{{\underline}{k}}\mapsto t^{{\underline}{k}\hspace{1pt}\cdot\hspace{1pt}{\underline}{\ell}}z^{{\underline}{k}} \;,$$ hence $z^{{\underline}{k}}$ belongs to ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$ if and only if ${\underline}{k}\cdot{\underline}{\ell}=0$.

The number of non-zero components of ${\underline}{k}$ is called the *length* of $z^{{\underline}{k}}$.

With $\ell_{i:j}:=\ell_i/{\mathrm}{gcd}(\ell_i,\ell_j)$, consider the invariant elements $$\label{eq:conj}
\xi_{i,j}:=(z_i^*)^{\ell_{j:i}}z_{\smash[t]{j}}^{\ell_{i:j}}\;,\qquad\forall\;i,j=0,\ldots,n \;.\vspace{5pt}$$

Any length $2$ element $z^{{\underline}{k}}\in{\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$ is a power of some $\xi_{i,j}$ as in .

Let $i\neq j$. A length $2$ element $z^{{\underline}{k}}=z_i^{\smash[b]{k_i}}z_{j}^{\smash[b]{k_j}}$ is invariant under the coaction if and only if $k_i\ell_{i:j}+k_j\ell_{j:i}=0$. Since $\ell_{i:j}$ and $\ell_{j:i}$ 