A Quiver is a directed graph. In other words, it is a collection of a bunch of dots (or sometimes numbers) with directed arrows between them. We call the dots vertices and the arrows, well arrows. There is no restriction on the number of vertices and arrows. You can have finitely many or even infinitely many vertices and arrows! However, we only focus on finite quivers, which means we have a finite number of vertices and arrows. Here’s an example of such an object. [Example graphic not included]
If you place your finger on a vertex, a path is the collection of arrows you take as you move your finger along the arrows. You can do nothing, which we call a stationary path. You can be creative and take as long or short of a path as you want. But the rule is, you have to follow the direction of the arrows. You can also set some of the paths “equal to zero” and put other relations among them. The collection of all of these paths form an object called a bound quiver algebra over a field K.
The objects we are interested in are called quiver representations. You take a quiver and place vector spaces at the vertices and matrices on the arrows. The resulting object is called a quiver representation. There are two ways one can study these quiver representations, algebraically or geometrically.
From the algebraic perspective, a quiver representation corresponds to a very well known algebraic object called a module over an algebra. These modules have building blocks called indecomposable modules which you build all of your possible modules from. The algebras can have finitely or infinitely many building blocks.
From a geometric perspective, you first fix the vector spaces at the vertices. This “fixing” of the vector spaces is called the dimension vector. The matrices on the arrows are allowed to vary but they have to satisfy the relations of the bound quiver algebra. The collection of all possible matrices on the arrows is called the affine module variety of a fixed dimension vector. Long story short, this object can be described by the solutions of polynomials of multiple variables and you can use geometric tools to study quiver representations.
We are interested in algebras that have module varieties with a special property. For each dimension vector, the module variety must have a dense quiver representation in each of its irreducible components. A dense orbit can be thought of as an object that “fills up” the space it lives in. The types of algebras with such properties are called dense orbit algebras or algebras with the dense orbit property.
We can then relate both the geometric and algebraic perspectives together and ask, which algebras are dense orbit algebras? If we have finitely many building blocks, then it is very well know that the associated algebra is of dense orbit algebra. If we have a dense type algebra, does it mean we have finitely many building blocks? The answer is no! In some cases, this is true but not always. Our long term goal is to find more of these dense orbit algebras that have infinitely many building blocks to hopefully classify them.