Classification of the Manin triples for complex reductive lie algebras

We study real and complex Manin triples for a complex reductive Lie algebra g. First, we generalize results of E. Karolinsky (1996, Math. Phys. Anal. Geom 3, 545-563; 1999, Preprint math.QA.9901073) on the classification of Lagrangian subalgebras. Then we show that, if g is noncommutative, one can attach to each Manin triple in g another one for a strictly smaller reductive complex Lie subalgebra of g. This gives a powerful tool for induction. Then we classify complex Marlin triples in terms of what we call generalized Belavin-Drinfeld data. This generalizes, by other methods, the classification of A. Belavin and V. G. Drinfeld of certain r-matrices, i.e., the solutions of modified triangle equations for constants (cf. A. Belavin and V. G. Drinfeld, "Triangle Equations and Simple Lie Algebras," Mathematical Physics Reviews, Vol. 4, pp. 93-165, Harwood Academic, Chur, 1984, Theorem 6.1). We get also results for real Martin triples. In passing, we retrieve a result of A. Panov (1999, Preprint math.QA.9904156) which classifies certain Lie bialgebra structures on a real simple Lie algebra. (C) 2001 Elsevier Science.