Why are (representations of ) quivers such a big deal?
Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, Gabriel gives a complete classification of quivers of finite type using just five Dynkin diagrams.
Although these are both deep and surprising, but I am not sure why quivers deserve so much attention. The only potential application I can think of (although highly unlikely to be true) that they might be useful to answer certain questions in category theory since the notion of quivers are similar to categories, and a representation is very much like a functor from a quiver to some $\mathcal{k}$-$\operatorname{vect}$.
So I wonder whether someone can give a hint why quivers deserve so much attention? Do they naturally show up in problems? And do representations of quivers really help to solve these problems?
Solution 1:
Morita Equivalence This is a supplement to the aspect of quiver representations mentioned in Alistair's answer. Every associative finite dimensional $k$-algebra $A$ is Morita equivalent to a path algebra $kQ/I$ (this is another Gabriel's theorem). In particular, you have a very nice equivalence (so nice that the functors giving such equivalence are given by tensoring finitely-generated projective bimodules) of abelian categories $A\text{-Mod}$ and $kQ/I\text{-Mod}$. So basically (almost) everything you want to know about representations of $A$ ($A$-modules) can be learnt from studying $kQ/I$-modules. And studying quiver representation is arguably much easier because path algebras are basic, meaning all simple modules are one-dimensional. This is equivalent to saying all projective indecomposable are only of multiplicity $1$ in $kQ/I$, one can say that this makes the homological behaviour of the modules much easier to study. In particular, many things can be done combinatorially.
Auslander-Reiten Theory This part does not relate directly to "why quiver representation", but to "why quivers". It turns out we can treat abelian categories (or in fact functorially finite categories) pretty much the same way as we treat an algebra: you can talk about irreducible maps. In particular, there is a combinatorial gadget called the Auslander-Reiten quiver, where vertices are in correspondence with indecomposable $A$-module and arrows are given by irreducible maps. In such a way, one can "visualise" the category nicely with a very nice form of quivers. And surprisingly, the form of quiver appearing in this construction also follows Dynkin classification.
Cluster Theory One of the most exciting developments in representation theory in the last decade is the cluster theory introduced by Fomin and Zelevinsky. The centre of the theory is an algebra called the cluster algebra, which is some sort of dual to the picture we see in the theory of Lie algebras (I do not know the exact argument to this). But the algebraic setting for clusters theory is pretty difficult to work with sometimes, and it turns out we can use an operation on quivers called a quiver mutation to substitute basis elements of the cluster algebra. Now people "categorify" this setting (which is in fact an incarnation of the Auslander-Reiten theory I mentioned above) and found out that we can use the derived category of the module category of the path algebra $kQ$ to study properties of cluster algebras.
Hall Algebras There is one construction of algebra called the Hall algebra of an abelian category. Ringel proved the amazing theorem in the 90s that if you take a Dynkin quiver $Q$ and consider the module category $kQ\text{-mod}$, then take the Hall algebra $H_Q$ of $kQ\text{-mod}$, it turns out $H_Q$ is isomorphic as an Hopf algebra to one half of the quantum group of the Lie algebra of type $Q$. I.e. studying quiver representations allow us to dig out more unknown properties of quantum group.
Quiver Varieties & Geometric Representation Theory (This is the impression I have got. Please correct me if I am wrong) If you recall from the proof of Gabriel's theorem on the classification of finite-type (unquotiented) path algebras, you will see there is some action of general linear group on the quiver. Nakajima extended this idea and developed a whole new approach for doing representation theory via geometric methods, using the so called Nakajima quiver variety.
Solution 2:
I would say that one of the reasons (and probably the main motivation for the work of Gabriel that you mention) is that the representation theory of quivers is intimately related to the representation theory of finite-dimensional associative algebras. If $A$ is a finite-dimensional algebra over some field $k$, then the category of representations of $A$ is equivalent to the category of representations of the algebra $kQ/I$ for some quiver $Q$ and some two-sided ideal $I$ of $kQ$. Here $kQ$ is the path algebra of the quiver. Representations of the path algebra are equivalent to representations of the quiver. Thus, representations of $kQ/I$ are equivalent to representations of the quiver that are killed by certain paths (i.e. the composition of maps along certain paths is zero).