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Journal of Lie Theory 22 (2012), No. 3, 741--756
Copyright Heldermann Verlag 2012

Abelian Ideals of Maximal Dimension for Solvable Lie Algebras

Dietrich Burde
Fakultät für Mathematik, Universität Wien, Nordbergstr. 15, 1090 Wien, Austria

Manuel Ceballos
Dep. Geometria y Topologia, Universidad de Sevilla, Sevilla, Spain

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 ≤ 7. 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.

Keywords: Abelian ideals, abelian subalgebras, degenerations.

MSC: 17B30, 17D25

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