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.