In the 1960's, W. Leavitt studied a class of universal algebras which do not have a well-defined rank, i.e., algebras $L$ for which $L^m\cong L^n$ as $L$-modules with $m\lt n$, later known as the Leavitt algebra $L(m,n)$. In two simultaneuous but independent studies by G. Abrams and G. Pino, and P. Ara et al., an algebra arising from a directed graph $E$ and a field $K$ has been introduced called the Leavitt path algebra $L_K(E)$. This algebra turned out to be the generalization of $L(1,n)$. In fact, $L(1,n)\cong L_K(R_n)$ where $R_n$ is the graph having one vertex and $n$ loops.

In 2013, R. Hazrat formulated the Graded Classification Conjecture for Leavitt path algebras which claims that the so-called talented monoid is a graded Morita invariant for the class of Leavitt path algebras. In this talk, we will see Leavitt Path algebra modules as a special case of quiver representation with relations and how it will take a role in proving the conjecture.

More concretely, we will provide evidences and a confirmation in the finite-dimensional case of the conjecture.

This is a compilation of joint works with Wolfgang Bock, Roozbeh Hazrat, and Jocelyn Vilela.