Journal of Lie Theory 23 (2013), No. 1, 217--228
Copyright Heldermann Verlag 2013
Algèbres de Lie 2-Nilpotentes et Structures Symplectiques
Dept. of Mathematics, Faculty of Mathematics and Informatic, LPMA, M'Sila University, Algeria
2-step nilpotent Lie algebras are finite dimensional Lie algebras A over a field with [[x,y],z] = 0 for all x,y,z of A. Each of them is a direct product of an abelian ideal and an ideal B with DB = ZB and we get three numerical invariants r = dim I, s = dim DA = dim DB. To classify these algebras, it is enough to consider only the case r = 0 (or DA = ZA) and we call (t,s) the type of A. In the article "Algèbre de Lie métabéliennes" [Ann. Faculté des Sciences Toulouse II (1980) 93--100] Ph. Revoy used the Scheuneman invariant [see J. Scheuneman, Two-step nilpotent Lie algebras, J. of Algebra 7 (1967) 152--159] to describe some of these; the aim of this paper is to complete and to make precise our earlier results, especially the case of s=2 or 3.
We study symplectic structures on 2-step nilpotent Lie algebras and we show that they are rarely symplectic algebras. Finally, symplectic Lie algebras play a role in superconformal field theories [see S. E. Parkhomenko, Quasi-Frobenius Lie algebra constructions of N = 4 superconformal field theories, Mod. Phys. Lett. A 11 (1996) 445--461] and have been studied in connection with rational solutions of the classical Yang-Baxter equation [see A. Stolin, Rational solutions of the classical Yang-Baxter equation and quasi Frobenius Lie algebras, J. Pure Appl. Algebra 137 (1999) 285--293].
Keywords: Two-step nilpotent Lie algebras, symplectic Lie algebras.
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