A characteristic property of simple Lie algebras
A finite-dimensional complex Lie algebra \({\mathfrak g}\) is called metrizable if on \({\mathfrak g}\) there is a nondegenerate symmetric bilinear form \(B\) invariant under the associated representation \(B([z,x],y)+B(x,[z,y])=0.\) The author studies certain properties of the decomposition of an arbitrary metrizable Lie algebra, which is not nilpotent, relative to a fixed Cartan subalgebra. The investigation follows, in the main, the standard scheme used in the theory of semisimple Lie algebras, although parts of the results and proofs of this theory are invalid or require refinements when metrizable Lie algebras are considered. The main results of this paper are : Theorem. Let \({\mathfrak g}\) be an arbitrary metrizable Lie algebra, R a system of nonzero roots relative to a fixed Cartan subalgebra, and \(R=R_ 1\cup R_ 0\) a decomposition of roots into so-called nonisotropic and isotropic parts. Then \({\mathfrak g}\) can be decomposed into the orthogonal direct sum \({\mathfrak g}={\mathfrak g}_ 1\oplus {\mathfrak g}_ 0\) of ideals \({\mathfrak g}_ 1\) and \({\mathfrak g}_ 0\), and \(R_ 1\), \(R_ 2\) are systems of nonzero roots for \({\mathfrak g}_ 1\) and \({\mathfrak g}_ 0\), respectively. The ideal \({\mathfrak g}_ 1\) is semisimple. Corollary 1. If all the roots of a nonsplittable metrizable Lie algebra \({\mathfrak g}\) are nonisotropic, then \({\mathfrak g}\) is simple. - Corollary 2. If a nonsplittable metrizable Lie algebra is not simple, then all its roots are isotropic. - Proofs are omitted.
scientific article; zbMATH DE number 3933282