Sur les triples de Manin pour les alg\`ebres de Lie r\'eductives complexes

Abstract

We study Manin triples for a reductive Lie algebra, $\g$. First, we generalize results of E. Karolinsky, on the classification of Lagrangian subalgebras (cf. KAROLINSKY E., {\em A Classification of Poisson homogeneous spaces of a compact Poisson Lie group}, Dokl. Ak. Nauk, 359 (1998), 13-15). Then we show that, if $\g$ is non commutative, one can attach, to each Manin triple in $\g$, an other one for a strictly smaller reductive complex Lie subalgebra of $\g$. We study also the inverse process.