What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl\_n$ be the general linear Lie algebra. Classify all Lie cobrackets $\delta: gl\_n \to \Lambda^2 gl\_n$. Any help will be greatly appreciated!


Edit: it seems that the case that $g$ is a semisimple Lie algebra is done by Belavin and Drinfeld. 



Classifications of Lie bialgebras
**Semisimple case:** Belavin-Drinfeld result does **not** classify all lie bialgebras on a (finite-dim.) semisimple complex Lie algebra, but only so called *quasi-triangular Lie bialgebras* (those having a non skew-symmetric r-matrix which satisfies CYBE). If one moves from the quasi-triangular case to the triangular case it can be easily shown that classification of triangular Lie bialgebra structures contains, as a subproblem, that of determining all Frobenius Lie subalgebras, ie. Lie subalgebras such that there exists a linear functional $l$ on them so that the bilinear form $l([X,Y])$ is nondegenerate. Semisimple Lie algebras are not Frobenius but may contain many (non semisimple) Frobenius Lie subalgebras. Unfortunately (and I think Alexander Chervov is referring to this fact) the classification of all Lie subalgebras of a given Lie algebra is a *wild* problem. See [When is a classification problem "wild"?](https://mathoverflow.net/questions/10481/when-is-a-classification-problem-wild)


You can find a neat explanation of all this in Korogodski-Soibelman *Algebras of functions on Quantum Groups Part I*, Math. Surv and Monographs 36, AMS 1998.


**General case:** in the general, not even semisimple case, still there are a bunch of interesting results. Lie bialgebra structures on 3-dim Lie algebras have been classified (in fact this result is periodically republished...). I would credit for this Xavier Gomez, Journ. Math. Phys. 41 (2000). 


Lie bialgebras were also classified on some specific Lie algebras like Heisenberg-Lie (Christian Ohn in dimension 3 and Andrè Diatta in general, I think - don't know if this last result were published apart from his PhD thesis). 


I've seen results also on classifying all Lie bialgebra structures on 4-dim Lie algebras but the list is huge and not very illuminating.


