Solvable complete Lie algebras. I
Suppose that \(N\) is a nilpotent Lie algebra, the maximal torus \(H\) of \(N\) is defined to be the maximal Abelian subalgebra consisting of the semisimple derivations of \(\text{Der} N\). The rank of \(N\) is defined to be \(\dim H\). In general, \(\dim H\leq \dim N/[N,N]\). If \(\dim H= \dim N/[N,N]\), \(N\) is said to be a nilpotent Lie algebra of maximal rank. In this letter the authors give the following two facts: 1) Any solvable complete Lie algebra \(L\) has a decomposition \(L=H \dotplus N\) such that \(N\) is the nilpotent radical of \(L\) and \(ad H |_N (\cong H)\) is a maximal torus on \(N\). The authors call \(L\) a solvable Lie algebra of maximal rank if \(\dim H= \dim N/[N,N]\), and \(L\) a non-maximal rank if \(\dim H< \dim N/[N,N]\). They point out the existence of solvable complete Lie algebras of non-maximal rank. 2) Let \(N\) be a nilpotent Lie algebra of maximal rank and \(H\) a maximal torus on \(N\). Then \(L= H \dotplus N\) is complete. All of the conclusions of this letter are asserted without any proof.
scientific article; zbMATH DE number 940627