Abelian ideals of maximal dimension for solvable Lie algebras
The authors compare the maximal dimensions of abelian subalgebras and abelian ideals for finite dimensional Lie algebras \(L\). Let \(\alpha(L)\) be the former and \(\beta(L)\) be the latter. There is a large literature on these invariants and the authors present an excellent review. In particular they list \(\alpha(L)\) for the complex simple Lie algebras and note the remarkable fact that \(\alpha(L)=\beta(B)\) where \(B\) is a Borel subalgebra of \(L\). They show another remarkable fact: \(\alpha(L)=\beta(L)\) for any solvable Lie algebra over an algebraically closed field of characteristic 0. When the abelian subalgebra has codimension 1, the ideal is constructed explicitly. Also, \(\alpha(L)\) is determined for each indecomposable nilpotent \(L\) of dimension \(\leq 7\), the case of dimension 7 being new.
scientific article; zbMATH DE number 6072205