Algèbres de lie 2-nilpotentes et structures symplectiques

摘要

2-step nilpotent Lie algebras are finite dimensional Lie algebras A over a field with [[x;y]; z] = 0 for all x; y; z in 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 = dimDA = dimDB. 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 métabé liennes, Ann. Faculté des Sciences Toulouse II (1980), 93{100," Ph. Revoy used the Scheuneman invariant (see Scheuneman, J., 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 al- gebras play a role in superconformal field theories (see Parkhomenko, S. E., 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 Stolin, A., Rational solutions of the classical Yang-Baxter equation and quasi Frobenius Lie algebras, J. Pure Appl. Algebra 137 (1999), 285{293).