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.
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"?
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.
Sorry for the self publicity, but for the especific example of $gl_n(k)$, you can view it as $gl_n(k)\cong sl_n(k)\times k$ and we did some work for trivial central extensions that allow you to produce, in particular, all Lie bialgebra extensions on $gl_n$ from data in $sl_n$, independently of the fact that they are comming from Belavin-Drinfel'd classification (they did the complex factorizable case) or not. If you denote $t$ is a generator of the abelian factor in $gl_n$, then all Lie bialgebra structures on $gl_n$ are of the form $$\delta (t)=0$$ $$\delta (x)=\delta_{sl_n}(x)+D(x)\wedge t$$ ($x\in sl_n$) where $\delta_{sl_n}$ is a Lie bialgebra structure on $sl_n$ and $D$ is a derivation of $sl_n$ that is also a coderivation w.r.t. $\delta_{sl_n}$. Equivalently, $$D(x)=[H,x]$$ for some $H\in sl_n$ such that $\delta_{sl_n}(H)=0$. Also equivalently, if $\delta_{sl_n}(x)=ad_x(r)$, then $H$ must satisfy $ad_H(r)=0$.
For example, you can take $\delta_{sl_n}\equiv 0$ and pick $H_0\in sl_n$, then $$\delta(t)=0$$ $$\delta(x)=[H_0,x]\wedge t$$ is a Lie bialgebra structure on $gl_n$.
If $\delta_{sl_n}$ is one of the structures comming from Belavin-Drinfel'd list, then $D=[H,-]$ with $H$ in the same Cartan subalgebra from Belavin-Drinfel'd statement, with some additional conditions. You can take a look at https://arxiv.org/pdf/1110.1072.pdf
We also did some work when the underlying Lie algebra is 2-step nilpotent, generalizing the results for the Heisenberg. https://arxiv.org/pdf/1607.00300.pdf (by the way, the results for the Heisenberg were published, you can also find references there)
