an ideal of the algebra $\widehat{\AG}_s$ generated by commutation relations: $$\label{13-3.7}
[\asm,\asp]=\frac{1}{10} \left(\Poco - \Pooc\right);$$ $$\label{13-3.8}
[\mathbb{N}_1,\aspm]=\mp\aspm,\quad [\mathbb{N}_2,\aspm]=\pm\aspm,
\quad [\mathbb{N}_1,\mathbb{N}_2]=0;$$ $$\label{13-3.9}
[\aspm,\Pnck{n}{k}]=\frac{1}{\sqrt{10}}\left(\Pnck{n\pm 1}{k}-\Pnck{n}{k\pm 1}\right);$$ $$\label{13-3.10}
[\aspm,\Pnkc{n}{k}]=\frac{1}{\sqrt{10}}\left(\Pnkc{n\mp 1}{k}-\Pnkc{n}{k\pm 1}\right);$$ $$\label{13-3.11}
[\asmp,\Pnkm{k,\,l}{k-t}{l+t}]=
\frac{1}{\sqrt{10}}\left(\Pnkm{k,\,l}{k-t\pm 1}{l+t\mp 1}-
\Pnkm{k\mp 1,\,l\pm 1}{k-t}{l+t}\right);$$ $$\label{3a.12}
[\asmp,\Pnkm{k,\,l}{k+t}{l-t}]=
\frac{1}{\sqrt{10}}\left(\Pnkm{k,\,l}{k+t\pm 1}{l-t\mp 1}-
\Pnkm{k\mp 1,\,l\pm 1}{k+t}{l-t}\right);$$ $$\label{13-3.13}
[\Pnck{n}{k},\Pnkc{m}{l}]=\left(\Pnkm{l,\,m+k-l}{m+k-n}{n}-\Pnkm{n+l-k,\,k}{m}{n+l-m}\right);$$ $$\label{13-3.14}
[\Pnck{n}{k},\Pnck{m}{l}]=\Pnck{n}{l}-\Pnck{m}{k};$$ $$\label{13-3.15}
[\Pnkc{n}{k},\Pnkc{m}{l}]=\Pnkc{n}{l}-\Pnkc{m}{k};$$ $$\label{13-3.16}
[\mathbb{N}_1,\Pnck{n}{k}]=(k-n)\Pnck{n}{k},\qquad
[\mathbb{N}_2,\Pnck{n}{k}]=(n-k)\Pnck{n}{k};$$ $$\label{13-3.17}
[\mathbb{N}_1,\Pnkc{n}{k}]=(n-k)\Pnkc{n}{k},
\qquad [\mathbb{N}_2,\Pnkc{n}{k}]=(k-n)\Pnkc{n}{k};$$ $$\label{13-3.18}
[\Pnkm{k,\,l}{k-t}{l+t},\Pnck{n}{m}]=
\Pnkm{k+n-m,\,m}{k-t}{n+t}-\Pnkm{k,\,l}{k+l-n}{n};$$ $$\label{13-3.19}
[\Pnkm{k,\,l}{k+t}{l-t},\Pnck{n}{m}]=
\Pnkm{k+l-m,\,m}{k+t}{l-t}-\Pnkm{k,\,l}{k+l-n}{n};$$ $$\label{13-3.20}
[\Pnkm{k,\,l}{k+t}{l-t},\Pnkc{n}{m}]=
\Pnkm{m,\,k+l-m}{k+t}{l-t}-\Pnkm{k,\,l}{n}{k+l-n};$$ $$\label{13-3.21}
[\Pnkm{k,\,l}{k-t}{l+t},\Pnkc{n}{m}]=
\Pnkm{m,\,k+l-m}{k-t}{l+t}-\Pnkm{k,\,l}{n}{k+l-n};$$ $$\label{13-3.22}
[\Pnkm{k,\,l}{k\mp t}{l\pm t},\Pnkm{m,\,n}{m\mp s}{n\pm s}]=
\delta_{k,\,m\mp s}\delta_{l,\,n\pm s}\Pnkm{m,\,n}{k\mp t}{l\pm t}-
\delta_{m,\,k\mp t}\delta_{n,\,l\pm t}\Pnkm{k,\,l}{m\mp s}{n\pm s};$$ $$\label{13-3.23}
[\Pnkm{k,\,l}{k-t}{l+t},\Pnkm{m,\,n}{m+s}{n-s}]=
\delta_{k,\,m+s}\delta_{l,\,n-s}\Pnkm{m,\,n}{k-t}{l+t}-
\delta_{m,\,k-t}\delta_{n,\,l+t}\Pnkm{k,\,l}{m+s}{n-s};$$ $$\label{13-3.24}
[\mathbb{N}_1,\Pnkm{k,\,l}{k\mp t}{l\pm t}]=\mp t\Pnkm{k,\,l}{k\mp t}{l\pm t},\qquad
[\mathbb{N}_2,\Pnkm{k,\,l}{k\mp t}{l\pm t}]=\pm t\Pnkm{k,\,l}{k\mp t}{l\pm t}.$$

Finally, we define the algebra $\AG_s$ of sectorial oscillator as quotient algebra $$\label{13-3.25}
\AG_s=\widehat{\AG}_s/I_s.$$ In other words, algebra $\AG_s$ is Lie algebra with generators , , satisfy the commutation relations -. Clearly, that $\AG_s$ is a closure of the associative algebra generated by operators and relations .

Algebra of generalized radial oscillator
========================================

In this section, we define the infinite-dimensional Lie algebra $\AG_r$ of the radial oscillator just as for sectorial oscillator. Namely, we describe the (minimal) extension of an associative algebra determined by generators with commutation relations .

Using relations and , we obtain $$\label{13-4.1}
\arpm=\sqrt{\frac{2}{5}}\bigoplus_{k,\,l\geq0}\Pnkm{k,\,l}{k\pm 1}{l\pm 