1}.$$ From , and should be the following equality $$\label{13-4.2}
I_B(\frac{1}{2},\frac{1}{2})=\frac{1}{2}\left(\Pnck{0}{0}+\Pnkc{0}{0}-\Pnkm{0,\,0}{0}{0}\right).$$ Let’s define algebras $\AGte$ and $\AGto$ as associative algebras with generators $$\label{13-4.3}
\mathbb{I},\, \mathbb{N}_1,\, \mathbb{N}_2,\, \Pnkm{k,\,l}{k\pm t}{l\pm t}, \quad k,l,t\geq 0,$$ (where $k=2n,\,l=2m$ è $k=2n+1,\,l=2m+1$, respectively) satisfying commutation relations $$\label{13-4.4}
[\Pnkm{k,\,l}{k\pm t}{l\pm t},\,\Pnkm{m,\,n}{m\pm s}{n\pm s}]=
\delta_{k,\,m\pm s}\delta_{l,\,n\pm s}\Pnkm{m,\,n}{k\pm t}{l\pm t}-
\delta_{m,\,k\pm t}\delta_{n,\,l\pm t}\Pnkm{k,\,l}{m\pm s}{n\pm s};$$ $$\label{13-4.5}
[\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{13-4.6}
[\mathbb{N}_1,\Pnkm{k,\,l}{k\pm t}{l\pm t}]=\pm t \Pnkm{k,\,l}{k\pm t}{l\pm t},\quad
[\mathbb{N}_2,\Pnkm{k,\,l}{k\pm t}{l\pm t}]=\pm t \Pnkm{k,\,l}{k\pm t}{l\pm t},\quad
[\mathbb{N}_1,\mathbb{N}_2]=0.$$ Checking that the Jacobi identity true, we prove the following lemma:

The algebras $\AGte$ and $\AGto$ are (infinite-dimensional) Lie algebras.

Using the decomposition , we introduce algebra $\AGt$ $$\AGt=\AGte\bigoplus\AGto.$$ We define the Lie algebra $\AGh$ as an algebra obtained from $\AGt$ by adding of all formal series in generators , including operators , , and also the operators $$\label{13-4.7}
\Pnkm{k}{\pm}{l}=\bigoplus_{m\geq0}\Pnkm{m\pm k,\,k}{m\pm l}{l},\qquad
\Pnkm{k}{l}{\pm}=\bigoplus_{m\geq0}\Pnkm{k,\,m\pm k}{l}{m\pm l}.$$ The commutation relations for the algebra $\AGh$ are induced by the relations -. From è it follows that $$\Pnkm{k}{-}{l}=P^{(l)}_{\bullet,\,k},\quad \Pnkm{k}{l}{-}=P^{(l)}_{k,\,\bullet}.$$ The following commutation relations $$\label{13-4.8}
[\mathbb{N}_1,\mathbb{N}_2]=0,\quad
[\mathbb{N}_j,\Pnkm{k,\,l}{k\pm t}{l\pm t}]=\pm t\Pnkm{k,\,l}{k\pm t}{l\pm t},
\quad [\mathbb{N}_j,\arpm]=\pm\arpm, \quad j=1,2;$$ $$\begin{gathered}
[\mathbb{N}_1,\,\Pnkm{k}{+}{l}]=(l-k)\Pnkm{k}{+}{l},\quad
[\mathbb{N}_2,\,\Pnkm{k}{l}{+}]=(l-k)\Pnkm{k}{l}{+},\nn\\
[\mathbb{N}_1,\,\Pnkm{k}{l}{+}]=(l-k)\Pnkm{l}{k}{+},\quad
[\mathbb{N}_2,\,\Pnkm{k}{+}{l}]=(l-k)\Pnkm{k}{+}{l};\label{13-4.9}\end{gathered}$$ $$\label{13-4.10}
[\Pnkm{k,\,l}{k\pm t}{l\pm t},\Pnkm{m,\,n}{m\pm s}{n\pm s}]=
\delta_{k,\,m\pm s}\delta_{l,\,n\pm s}\Pnkm{m,\,n}{k\pm t}{l\pm t}-
\delta_{m,\,k\pm t}\delta_{n,\,l\pm t}\Pnkm{k,\,l}{m\pm s}{n\pm s};$$

