Classification des triples de Manin pour les algèbres de Lie réductives complexes

Let g be a finite dimensional, complex, reductive Lie algebra. One says that a symmetric, g-invariant, R(resp. C)-bilinear form on g is a Manin form if and only if its signature is (dimCg, dimCg) (resp. is non degenerate). Recall that a Manin-triple in g is a triple (B, i, i), where B is a real (resp. complex) Manin form and where i, i are real (resp. complex) Lie subalgebras of g, isotropic for B, and such that i + i = g. Then this is a direct sum and i, i are of real dimension equal to the complex dimension of g. Our goal is to classify all Manin-triples of g, up to conjugacy under the action on g of the connected, simply connected Lie group G with Lie algebra g, by induction on the rank of the derived algebra of g. One calls af-involution (resp. f-involution) of a complex semi-simple Lie algebra m, any R(resp. C)-linear involutive automorphism of m, σ, such that there exists : 1) an ideal m̃0 of m, which is reduced to zero for f-involutions, and a real form h̃0 of m̃0, 2) simple ideals of m, mj , m ′′ j , j = 1, . . . , p, 3) an isomorphism of complex Lie algebras, τj , between m ′ j and m ′′ j , j = 1, . . . , p, such that, denoting by hj := {(X, τj(X))|X ∈ m ′ j}, and by h the fixed point set of σ, one has :