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

TL;DR

This paper classifies complex and real Manin triples for complex reductive Lie algebras, extending previous work, introducing a reduction method, and generalizing Belavin-Drinfeld classification of R-matrices.

Contribution

It generalizes classification of Lagrangian subalgebras, introduces a reduction technique for Manin triples, and extends Belavin-Drinfeld classification to broader contexts.

Findings

Classified complex Manin triples using generalized Belavin-Drinfeld data.

Developed a reduction method attaching smaller subalgebra triples.

Extended classification results to real Manin triples and Lie bialgebra structures.

Abstract

We study real and complex Manin triples for a complex reductive Lie algebra, $\g$. The first part includes, and extends to complex Manin triples, our earlier work [De]. First, we generalize results of E. Karolinsky, on the classification of Lagrangian subalgebras (cf.[K1], [K3]). Then we show that, if $\g$ is non commutative, one can attach, to each Manin triple in $\g$, another one for a strictly smaller reductive complex Lie subalgebra of $\g$. This gives a powerful tool for induction. Then we classify complex Manin triples, in terms of what we call generalized Belavin-Drinfeld data. In particular this generalizes, by other methods, the classification of A. Belavin and G. Drinfeld of certain $R$-matrices, i.e. the solutions of modified triangle equations for constants (cf [BD], Theorem 6.1). We get also results for real Manin triples. In passing, one retrieves a result of A. Panov [P1] which classifies certain Lie bialgebras structures on a real simple Lie algebra.