Central extensions of Lie algebras graded by finite root systems (Q1974131)

This work completes the classification of Lie algebras \(L\) over a field of characteristic zero that are graded by a root system \(\Delta\) where \(\Delta\) is the root system of a finite-dimensional split simple Lie algebra \(\mathfrak g\) relative to a split Cartan subalgebra. As a \(\mathfrak g\)-module such an algebra is a direct sum of modules isomorphic to \(\mathfrak g\), modules isomorphic to the irreducible \(\mathfrak g\)-module \(V\) whose highest weight is the highest short root, and one-dimensional \(\mathfrak g\)-modules, i.e. \(L = ({\mathfrak g} \otimes A) \oplus (V\otimes B) \oplus {\mathcal D}\). Thus the classification is accomplished by determining \(A, B\), and \(\mathcal D\) and the multiplication between the summands. Since this classification has previously been achieved up to central extensions, this paper is devoted to determining the central extensions.