Connection between quivers and representations of Lie algebras Can anyone recommend a reference to study the connection between quiver theory and representation theory of Lie algebras?
Supposedly those two things have something to do with each other, with the notion of root system appearing in different places. However I apart from bearing the same name, I'm not sure what those definitions have to do with each other.

Gabriel's theorem is the one involving finite root systems in the representation theory of quivers.  It basically says that a connected quiver admits finitely many indecomposable representations if and only if it is an orientation of a Dynkin diagram of type ADE, and gives a bijection between the set of indecomposable representations and the set of positive roots of the associated root system.
As for references, I recommend Gelfand, Berenstein and Ponomarev's paper Coxeter functors and Gabriel's theorem for a nice proof of the theorem.  Of course, there are more modern texts giving an account of the theorem, for instance Assem, Simson and Skowroński's book Elements of the Representation Theory of Associative Algebras I Techniques of Representation Theory.
From there, there are many other links between Lie algebras and representation theory of quivers.  I will just mention a theorem of Ringel, realizing the quantum enveloping algebra of the positive part of a Lie algebra as the Ringel-Hall algebra of the associated quiver.
