Manin Triples of Real Simple Lie Algebras. Part 2

We complete the study of Manin triples of real simple Lie algebras. In the Part 2 of the article we classify the Manin triples (g(R), W, g(R) ⊕ g(R)) (case 2 of the doubles) up to weak and gauge equivalence. First we recall the main definitions of the Part 1. 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. 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. We study Manin triples as follows. First for given g we find all the doubles d and describe all forms Q(·, ·) on d such that g is an isotropic subspace with respect to this form. Then fixing d and Q we study complementary subalgebras W such that (g, W, d) forms a Manin triple. We are going the classify Manin triples of real simple Lie algebras up to weak and gauge equivalence. It is known that the double d of complex simple Lie algebra g is isomorphic to one of following Lie algebras g ⊗ A i (C) , where A 1 (C) = C[t]/t 2 , A 2 (C) = C[t]/(t 2 − 1/4). In Part 1 of this paper we study the action of conjugation σ of C/R on complex doubles and obtain that the double d(R) of real simple Lie algebra g(R) is isomorphic to one of following algebras g ⊗ A i (R), where A 1 (R) = R[t]/t 2 , A 2 (R) = R[t]/(t 2 − 1/4), A 3 (R) = C. So, according to the double, there exist 3 types of Manin triples. In Part 1 we consider Manin …