The computation of Abelian subalgebras in low-dimensional solvable Lie algebras

Abstract

The main goal of this paper is to compute the maximal abelian dimension of each solvable nondecomposable Lie algebra of dimension less than 7. To do it, we apply an algorithmic method which goes ruling out non-valid maximal abelian dimensions until obtaining its exact value. Based on Mubarakzyanov and Turkowsky’s classical classifications of solvable Lie algebras (see [13] G.M. Mubarakzyanov: Classification of real structures of Lie algebras of fifth order. Izv. Vyss. Ucebn. Zaved. Matematika 3:34, 1963, pp. 99-106. and [19] P. Turkowski: Solvable Lie algebras of dimension six. J. Math. Phys. 31, 1990, pp. 1344-1350) and the classification of 6-dimensional nilpotent Lie algebras by Goze and Khakimdjanov [7] M. Goze and Y. Khakimdjanov: Nilpotent and solvable Lie algebras. In M. Hazewinkel (ed.): Handbook of Algebra Vol 2. Elsevier, Amsterdam, 2000, pp. 615–664, we have explicitly computed the maximal abelian dimension for the algebras given in those classifications.