Classification des triples de Manin pour les algèbres de Lie réductives complexes: Avec un appendice de Guillaume Macey

Abstract

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 Manin 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 Manin 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.