Classification of finite-dimensional solvable Lie algebras with nondegenerate invariant bilinear forms
In the paper under review one investigates the classification of finite-dimensional complex solvable Lie algebras that allow for a nondegenerate symmetric invariant bilinear form \((\cdot\mid\cdot)\colon{\mathfrak g}\times{\mathfrak g}\to{\mathbb C}\). One proceeds by taking into account root space decompositions corresponding to a maximal abelian subalgebra \({\mathfrak h}\) of \({\mathfrak g}\) with the property that the linear mapping \(\text{ad}_{\mathfrak g}x\colon{\mathfrak g}\to{\mathfrak g}\) is a semisimple derivation for all \(x\in{\mathfrak h}\). Specifically, one looks at minimal generating subsets of \({\mathfrak g}\) consisting of root vectors with respect to the aforementioned root space decomposition and one singles out three possible situations according to certain orthogonality relations in terms of the bilinear form \((\cdot\mid\cdot)\) on such a generating set. In each of these three situations, one provides information on the structure of the solvable Lie algebra \({\mathfrak g}\) under consideration.
scientific article; zbMATH DE number 5148859