Abstract
This paper deals with the maximal abelian dimension of a Lie algebra, that is, the maximal value for the dimensions of its abelian Lie subalgebras. Indeed, we compute the maximal abelian dimension for every nilpotent Lie algebra of dimension less than 7 and for the Heisenberg algebra \(\mathfrak{H}_k\), with \(k\in\mathbb{N}\). In this way, an algorithmic procedure is introduced and applied to compute the maximal abelian dimension for any arbitrary nilpotent Lie algebra with an arbitrary dimension. The maximal abelian dimension is also given for some general families of nilpotent Lie algebras.
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Benjumea, J.C., Núñez, J., Tenorio, A.F.: The maximal abelian dimension of linear algebras formed by strictly upper triangular matrices. Theor. Math. Phys. 152, 1225–1233 (2007)
Frè, P.: Solvable Lie algebras, BPS black holes and supergravity gaugings. Fortschr. Phys., 47, 173–181 (1999)
Frè, P., Gili, V., Gargiulo, F., Sorin, A., Rulik, H., Trigiante, M.: Cosmological backgrounds of superstring theory and solvable algebras: oxidation and branes. Nucl. Phys. B 685, 3–64 (2004)
Goze, M., Khakimdjanov, Y.: Nilpotent Lie Algebras. Kluwer, Dordretch (1996)
Grunewald, F., O’Halloran, J.: Varieties of nilpotent lie algebras of dimension less than six. J. Algebra 112, 315–325 (1988)
Heber, J.: Non-compact homogeneous Einstein spaces. Inv. Math. 133, 279–352 (1998)
Hervik, S.: Einstein metrics: homogeneus solvmanifolds, generalised Heisenberg groups and black holes. J. Geom. Phys. 52, 298–312 (2004)
Petrov, A.Z.: Einstein Spaces. Pergamon, Oxford (1969)
Schimming, R.: On constant solutions of the Yang–Mills equations. Arch. Math. (Brno) 24, 65–73 (1988)
Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2003)
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Benjumea, J.C., Núñez, J. & Tenorio, Á.F. Maximal Abelian Dimensions in Some Families of Nilpotent Lie Algebras. Algebr Represent Theor 15, 697–713 (2012). https://doi.org/10.1007/s10468-010-9260-4
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DOI: https://doi.org/10.1007/s10468-010-9260-4