Triples of Real Simple Lie Algebras

The article is devoted to the problem of classification of Manin triples up to weak and gauge equivalence. The case of complex simple Lie algebras can be obtained by papers of A.Belavin, V.Drinfel'd, M.Semenov-Tian-Shanskii. Studing the action of conjugaton on complex Manin triples, we get the list of real doubles. There exists three types of the doubles. We classify all ad-invariant forms on the double compatible with bialgebra structure. We classify the Manin triples (g(R), W, g(C)) (case 3 of the doubles) up to weak and gauge equivalence. The problem of classificaton of the Lie-Poisson brackets on Lie groups leads to the notion of Manin triple. Definition 1. Let g 1 , g 2 , d be Lie algebras over a field K and let Q be a symmetric nondegenerate bilinear form on d. A triple (g 1 , g 2 , d) is called a Manin triple if Q(x, y) is ad-invariant and d is a direct sum of maximum isotropic subspaces g 1 , g 2. If g = g 1 , then g 2 can be identifined with g *. The algebra d is called a double of g. The Lie algebra structure on g * induces a Lie coalgebra structure on g and these structures are compatible. In this case we say that g is a Lie bialgebra. Definition 2. We say that two Manin triples (g, W, d) and (g, W ′ , d) are weak equivalent if there exists an element a in the adjoint group D of the double d such that W ′ = Ad a (W). Definition 3. We say that two Manin triples (g, W, d) and (g, W ′ , d) are gauge equivalent if there exists an element a in the adjoint group D of double d such that W ′ = Ad a (W) and Ad a (g) = g. Every two gauge equivalent Manin triples are weak equivalent. One set up the problem of classification of all Manin triples up to weak and gauge equivalence in terms of Lie structure of g. The paper is organized as follows. In the section 1 we recall the classification of Manin triples of simple complex Lie algebras. In other sections, we study real Manin triples. The complexification of a real Manin triple is a complex Manin triple. We treat the extentions of conjugation σ of the complex Lie algebra …