Abelian ideals of maximal dimension for solvable Lie algebras

TL;DR

This paper investigates the maximal dimensions of abelian subalgebras and ideals in solvable Lie algebras, showing their equality in characteristic zero and computing these invariants for small nilpotent cases.

Contribution

It proves the equality of these dimensions for solvable Lie algebras over algebraically closed fields of characteristic zero and explicitly constructs abelian ideals of codimension 2 in nilpotent cases.

Findings

Maximal dimensions of abelian subalgebras and ideals coincide for solvable Lie algebras.

Computed invariants for all complex nilpotent Lie algebras of dimension less than 8.

Constructed explicit abelian ideals of codimension 2 in nilpotent Lie algebras.

Abstract

We compare the maximal dimension of abelian subalgebras and the maximal dimension of abelian ideals for finite-dimensional Lie algebras. We show that these dimensions coincide for solvable Lie algebras over an algebraically closed field of characteristic zero. We compute this invariant for all complex nilpotent Lie algebras of dimension n less than 8. Furthermore we study the case where there exists an abelian subalgebra of codimension 2. Here we explicitly construct an abelian ideal of codimension 2 in case of nilpotent Lie algebras.