The root space decompositions of the quadratic Lie superalgebras
Let \(\mathfrak g=\mathfrak g_0\oplus \mathfrak g_1\) be a finite dimensional Lie superalgebra over a field of characteristic zero, \(H\) a Cartan subalgebra in \(\mathfrak g_0\). Suppose that there is given a supersymmetric even nondegenerate and \(\mathfrak g\)-invariant form on \(\mathfrak g\). Then \(\mathfrak g\) is basic classical iff the center of \(\mathfrak g\) is trivial and \(\text{ad} _{\mathfrak g}h\) is semisimple for all \(h\in H\). Suppose that the \(\mathfrak g_0\)-module \(\mathfrak g_1\) is semisimple and \(\Delta\) is the set of roots of \(\mathfrak g\) relative to \(H\). Then \(\mathfrak g\) is basic classical iff \(\Delta\) generates the dual space \(H^*\).
scientific article; zbMATH DE number 1428673