Representations of the two dimensional non-abelian Lie algebra I would like to know a complete description of the indecomposable representations of the two dimensional non-abelian Lie algebra over the complex numbers. The finite dimensional representations would also be interesting for me.
 A: A further comment on Pasha's answer, in community-wiki mode:
The Russian article cited by Pasha (which doesn't yet have free access online in Russian) is referred to later that year in the same journal:  E.A. Makedonskii, On wild and tame finite-dimensional Lie algebras. (Russian) Funktsional. Anal. i Prilozhen. 47 (2013), no. 4, 30–44; translation in Funct. Anal. Appl. 47 (2013), no. 4, 271–283.  
This later paper suggests an answer to the question raised, but only in the limited finite dimensional setting.  According to the MathSciNet review by V.V. Gorbatsevich, the only finite dimensional Lie algebras over an algebraically closed field of characteristic 0 for which the classification of finite dimensional representations is not a wild problem are the semisimple ones, or direct sums of semisimple ones and one-dimensional centers. 
A: I guess it follows as a very particular case, from the works about wild/tame dichotomy, representations of quivers, etc., that in characteristic zero the problem of classification of finite-dimensional indecomposable modules over a two-dimensional nonabelian Lie algebra, is wild. For example, it can be inferred from: 
V.L. Ostrovskyi and Yu.S. Samoilenko, On pairs of quadratically related operators, Funct. Anal. Appl. 47 (2013), N1, 67-71, 
where the problem of classification of pairs of linear operators A,B with [A,B] = polynomial (A) is reduced to some wild quiver problem.
On the other hand, the dimensions of restricted finite-dimensional indecmposable modules are bounded, as follows, for example, from:
R.D. Pollack, Restricted Lie algebras of bounded type, Bull. Amer. Math. Soc. 74 (1968), 326-331 DOI:10.1090/S0002-9904-1968-11943-3 ,
so they are probably, manageable. I do not know what the situation with arbitrary modules in positive characteristic is.
A: Thank you very much to Pasha Zusmanovich for pointing out the reference Ostrovskyi, Samoilenko. (by the way I would like to vote his answer but I do not know how to do it, it is my first visit to mathoverflow). I gave a look to their paper and I can write down their answer.
They never write down the definition of wild but I think they use the following definition (for me the definition is new so I hope that what I write is correct)
A k algebra A is wild if the category of finite dimensional representation of A contains a subcategory equivalent to the category of representations of the free associative algebras in two generators over k. 
If I understand correctly this implies that it contains a subcategory isomorphic to the category of the representations of any finitely generated k-algebra.
Also I do not understand if this definition of wild really implies that there is no hope for a classification. A priori it seems to me that a larger category could be simpler than the smaller one.
This is the argument of Ostrovskyi, Samoilenko:
Theorem: the category whose object are given by two vector spaces U,V and linear maps A:U-->U, B:U-->V, C:V-->V with AB=BC,  and A^n=C^n=0 with n \geq 5 (and the natural definition of morphism) is wild.
This is the category of representations of the algebra: with five generators A,B,C,I,J with I^2=I, J^2=J, AI=IA=A, JC=CJ=C, BI=B=JB, AB=BC, 
IB=BJ=IJ=JI=IC=CI=JA=AJ=0 and B invertible
They do not prove this theorem, they refer to a paper of Hoshino and Miyachi (Tsukuba J. of Math. 1988) and to a paper of Han (J.of algebra 2002). 
The claim that the representations of the algebra [x,y]=y is wild follows easily from the theorem: given a triple (A,B,C) consider representations of this algebra where x and y acts as 2x2 block matrices given by
x=(1+A 0 \ 0 C)   
y=(0 B \ 0 0)
Then [x,y]=y is equivalent to AB=BC and if the representation defined by x',y' is isomorphic to that defined by x,y is equivalent to require that the representation of the algebra of the theorem above associated to (A,B,C) is equivalent to that defined by (A',B',C')
I gave a look to the proof of Theorem, however in these papers they have a general approach which for me is not easy to follow (you end up to prove that the affine quiver of type E_7 is wild). Maybe would be possible for these specific algebra to produce a more direct argument (notice however that according to their table if you require A^3=C^3=0, instead that A^5=C^5=0, this should not be wild)
thank you again
Andrea
PS: My interest in the problem does not come from any particular reason. I just tried to give a look to the smallest possible example. I thought it would have been simpler. 
A: Quoth Dixmier, Enveloping algebras, p. xii: "But a deeper study reveals the existence of an enormous number of irreducible representations of [the 3-dimensional Heisenberg algebra]. It seems that these representations defy classification. A similar phenomenon exists for $\mathfrak g = \mathfrak{sl}(2)$, and most certainly for all non-commutative Lie algebras."
