Manin triples of real simple Lie algebras. Part 1

Abstract

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 the papers of A.Belavin, V.Drinfel'd, M.Semenov-Tian-Shanskii. Studing the action of conjugation on complex Manin triples, we get the list of real doubles. We classify all $ad$-invariant forms on the double compatible with bialgebra structure. We classify the Manin triples $(g(R),W,g(C))$ up to weak and gauge equivalence.