This work is done over an algebraically closed field $k$ (it can be assumed that $char(k)=0$). Now I consider all possible $k$-algebras $A$, up to isomorphism, with $rank_k(A)=4$ (i.e. $A$ has dimension $4$ as a vector space over $k$). But for finite dimensional algebras we have the following results:
$(1)$ Let $A$ be a finite-dimensional algebra over $k$. Then $A/J(A)$ is semisimple. Here $J(A)$ denotes the Jacobson radical of $A$.
$(2)$ Let $k$ be an algebraically closed field. If $B$ is a finite-dimensional semisimple $k$-algebra, then $B\cong \Pi_{i=1}^{s}M_{n_i}(k)$ for some natural numbers $n_i$, $1 ≤ i ≤ s$.
$(3)$ (Gabriel) The $k$-algebra $A$ is basic if and only if $A\cong kQ/I$ where $Q$ is a quiver and $I$ is an admissible ideal.
Now the assumption that $A$ is finite-dimensional (with $rank_k(A)=4$) implies that $A/J(A)$ is semisimple (by result $(1)$). Now result $(2)$ for $B=A/J(A)$, and $rank_k(A)=4$, implies that either $A/J(A)\cong M_2(k)$, or $A$ is basic.
If $A$ is basic, result $(3)$ implies that $A\cong kQ/I$ where $Q$ is a quiver and $I$ is an admissible ideal. The condition $rank_k(A)=4$ implies that the quiver $Q$ has at most $4$ vertices.
Now we consider cases for the number vertices and number of arrows. One case that seems more difficult is when $Q$ has only one vertex $V$ and two loops $x,y$ (starting and ending in $V$). The ideal $kQ_+$ is generated by $\{x,y\}$, so $kQ_+^m\subset I\subset kQ_+^2$ for some $m\geq 2$.
My Question: How to find all the isomorphism classes of $k$-algebras with rank 4 of the form $kQ/I$ in the case of the quiver described above? So, what are all possible ideals $I$ that suffice to do this classification.
One family of algebras comes from the ideal $I_{\lambda}=(x^2,y^2,yx-\lambda xy)$, for $\lambda\neq -1$ in $k$. For $\lambda\neq 0$ we get $kQ/I_{\lambda}\cong kQ/I_{\lambda'}$ iff $\lambda \lambda'=1$.
P.S. I am working with the paper "Finite representation type is open" by P. Gabriel; except the family above, Gabriel gives some other algebras coming from this quiver (a finite number of them).