negative.

Using , and decomposition , we have $$\label{13-3.2}
\aspm=\frac{1}{\sqrt{10}}\bigoplus_{m,\,n=0}^{\infty}P^{m,\,n}_{m\mp 1,\,n\pm 1}.$$ We introduce the operators $$\label{13-3.3}
P^{(n)}_{\bullet,\,k}=\bigoplus_{m=k}^{\infty}P^{m-k,\,k}_{m-n,\,n},
\qquad P^{(q)}_{p,\,\bullet}=\bigoplus_{l=p}^{\infty}P^{p,\,l-p}_{q,\,l-q},
\qquad k,n,q,p\geq0.$$ The relations and imply that $$P_1=\bigoplus_{N=0}^{\infty}P^{N,\,0}_{N,\,0}=P^{(0)}_{\bullet,\,0},
\qquad P_2=\bigoplus_{N=0}^{\infty}P^{0,\,N}_{0,\,N}=P^{(0)}_{0,\,\bullet}.$$ The action of all operators defined above shown in the following diagram.

![The action of operators , ](3_NEW){width="12.0cm"}

We will start building algebra $\mathfrak{A}_s$ of the sectorial oscillator with the definition of algebras $\mathfrak{A}^{(N)}_s$, for $N\geq0$, as associative algebras with generators $$\label{13-3.4}
\mathbb{I},\, \mathbb{N}_1,\, \mathbb{N}_2,\, P^{k,\,l}_{k\mp 1,\,l\pm 1},
\, P^{k,\,l}_{k,\,l},\quad
(k+l=N,\, k,l\geq0),$$ satisfying the commutation relations $$\begin{gathered}
[\mathbb{N}_1,\,\mathbb{N}_2]=0,
\quad [\mathbb{N}_1,\,P^{k,\,l}_{k,\,l}]=[\mathbb{N}_2,\,P^{k,\,l}_{k,\,l}]=0,\quad
[P^{k,\,l}_{k\,l},\,P^{m,\,n}_{m\,n}]=0,\nn\\
[\mathbb{N}_1,\,P^{k,\,l}_{k\mp 1,\,l\pm 1}]=\mp P^{k,\,l}_{k\mp 1,\,l\pm 1},\qquad
[\mathbb{N}_2,\,P^{k,\,l}_{k\mp 1,\,l\pm 1}]=
\pm P^{k,\,l}_{k\mp 1,\,l\pm 1},\label{13-3.5}\\
[P^{m,\,n}_{m\mp 1,\,n\pm 1},\,P^{k,\,l}_{k,\,l}]=
\delta_{m,\,k}\delta_{n,\,l}P^{k,\,l}_{m\mp 1,\,n\pm 1}
-\delta_{k,\,m\mp 1}\delta_{l,\,n\pm 1}P^{m,\,n}_{k,\,l},\nn\\
[P^{m,\,n}_{m-1,\,n+1},\,P^{k,\,l}_{k+1,\,l-1}]=\delta_{m,\,k+1}\delta_{n,\,l-1}
\left(P^{k,\,l}_{k,\,l}-P^{k+1,\,l-1}_{k+1,\,l-1}\right).
\nn\end{gathered}$$ It is easy to check the validity of the Jacobi identities. Therefore, the algebras $\mathfrak{A}^{(N)}_s,$ $N\geq0$, generated by operators satisfying the relations , are Lie algebras with dimension $3(N+2)$. Using the decomposition , we determine the infinite-dimensional Lie algebra $\widetilde{\mathfrak{A}}_s$ $$\AGts=\bigoplus_{N=0}^{\infty}\AG^{(N)}_s,$$ We define the Lie algebra $\widehat{\AG}_s$ as algebra, obtained from $\widetilde{\AG}_s$ by addition to generators of all formal series composed of these generators, including operators and . The commutation relations of the algebra $\widehat{\AG}_s$ are induced by the relations .

Let $I_s$ be 