The construction of solvable Lie algebras from equidimensional nilpotent algebras
The authors associate with each finite dimensional Lie algebra L with nilpotent derived algebra a unique nilpotent Lie algebra \(L^*\), called the nilpotent frame of L. The aim is to reduce the classification of solvable Lie algebras to the same problem for nilpotent Lie algebras. They begin by studying the decomposition of a matrix into its semisimple and nilpotent parts: \(a=s(a)+n(a)\). One obtains a linear epimorphism from a Cartan subalgebra of L onto a diagonal subalgebra A of the derivation algebra of L. This mapping is used to construct a monomorphism from L into the semi-direct sum of L and A, the image of which is the nilpotent frame, \(L^*\). The properties of \(L^*\) are investigated in this paper. In particular, over fields of characteristic 0, the nilpotent frame of L is unique up to an isomorphism induced by an inner automorphism of L. The reverse problem of starting with a nilpotent \(L^*\) and constructing a solvable Lie algebra with nilpotent frame \(L^*\) is also shown to have a positive solution.
scientific article; zbMATH DE number 4175203