Derivations and central extensions of finitely generated graded Lie algebras (Q1111681)

Let V be a module of a Lie algebra L over a field F. A linear mapping \(\phi\) : \(L\to V\) is called a derivation if \(\phi ([x,y])=x\cdot \phi (y)-y\cdot \phi (x)\). \(\phi\) is called inner if it is of the form \(x\mapsto x\cdot v\). If \(L=\oplus_{g\in G}L_ g\) is a G-graded Lie algebra where G is an abelian group, and \(V=\oplus_{g\in G}V_ g\) a G- graded module of L, then the space of derivations \(Der_ F(L,V)=\oplus_{g\in G}Der_ F(L,V)_ g\) where \(Der_ F(L,V)_ g=\{\phi \in Der_ F(L,V)\); \(\phi (L_ h)\subseteq V_{g+h}\), \(h\in G\}.\) The author establishes some general results concerning derivations of finitely generated graded Lie algebras which are applied to determine the derivations and central extensions of Kac-Moody Lie algebras \({\mathfrak g}(A)\) associated to an \(n\times n\) matrix \((a_{ij})\) of rank \(\ell\). Suppose Char F\(=0\) and \(a_{ii}\neq 0\), \(1\leq i\leq n\). The author determines the structure of \(Der_ F({\mathfrak g}(A),{\mathfrak g}(A))\) and \(Der_ F({\mathfrak g}(A),{\mathfrak g}(A)^*)\) and he proves that \[ \dim_ F H^ 1({\mathfrak g}(A),{\mathfrak g}(A))=(n-\ell)^ 2,\quad \dim_ F H^ 1({\mathfrak g}(A),{\mathfrak g}(A)^*)=n(n-\ell)^ 2,\quad \dim_ F H^ 2({\mathfrak g}(A),F)=\left( \begin{matrix} n-\ell \\ 2\end{matrix} \right). \]