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.

  • $\begingroup$ Thank you, I will look into this. This seems a bit too advanced for me already though, as I am stillt puzzled what is the connection between "root system of a quiver with Dynkin diagram" and "root system for a Lie algebra". It's probably not very difficult to see, but for someone self-studying both topics it is difficult for me to figure out. $\endgroup$ – tortemath Jan 31 '16 at 17:08

There is another fascinating connection between ADE shaped quivers and an root systems of Lie algebras, namely the so-called geometric McKay correspondence. If you define the moduli space of representations of a Dynkin quiver with pre-projective relations, and quotient it (after throwing away "bad" points first) by the product of $GL_n$ groups acting via base change on each node, then you actually get a variety. This variety will be the Klein singularity associated with the ADE root system in question.

For example, take the quiver shaped like an $A_n$ Dynkin diagram, whereby links between nodes correspond to pairs of arrows, one going each way. Supplement this quiver with appropriate quadratic pre-projective relations. Now, the appropriately built moduli space of representations with dimension vector $d=(1,1,1, ..., 1)$ will actually be the $A_n$ singularity. By deforming the stability condition you use in order to make the GIT quotient, you will see a resolved singularity, with a chain of $P^1$'s connected in the shape of the $A_n$ Dynkin diagram.

Pretty cool stuff.


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