
Journal of Lie Theory 23 (2013), No. 1, 217228 Copyright Heldermann Verlag 2013 Algèbres de Lie 2Nilpotentes et Structures Symplectiques Noureddine Midoune Dept. of Mathematics, Faculty of Mathematics and Informatic, LPMA, M'Sila University, Algeria midounenour@yahoo.fr 2step 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) 93100] Ph. Revoy used the Scheuneman invariant [see J. Scheuneman, Twostep nilpotent Lie algebras, J. of Algebra 7 (1967) 152159] 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 2step 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, QuasiFrobenius Lie algebra constructions of N = 4 superconformal field theories, Mod. Phys. Lett. A 11 (1996) 445461] and have been studied in connection with rational solutions of the classical YangBaxter equation [see A. Stolin, Rational solutions of the classical YangBaxter equation and quasi Frobenius Lie algebras, J. Pure Appl. Algebra 137 (1999) 285293]. Keywords: Twostep nilpotent Lie algebras, symplectic Lie algebras. MSC: 17B30 [ Fulltextpdf (298 KB)] for subscribers only. 