In the 1960's. W. Leavitt studied universal algebras which do not satisfy the \emph{Invariant Basis Number Property (IBN)}. These are algebras that do not have a well-defined rank, that is, algebras for which $R^m\cong R^n$ ($m\neq n$) as $R$-modules which are later called the \emph{Leavitt algebras of module type} $(m,n)$. In 2005, the Leavitt algebra of type $(1,n)$ was found to be the so-called \emph{Leavitt path algebra} of a certain directed graph. 

In 2013, R. Hazrat formulated the \emph{Graded Classification Conjecture} for Leavitt path algebras. This claims that so-called \emph{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.