I've been refreshing my linear algebra, and this is a question of curiosity I have.
Let $U:=U_n(F)$ be the algebra of upper triangular $n\times n$ matrices over a field $F$. Is there a classification of all simple $U$-modules (up to isomorphism of course)? I've been researching around, but didn't find any relevant results on first look. I'd also be happy for a reference showing such classification if one exists. Thank you.
Best Answer
The Jacobson radical $J(R)$ of a ring $R$ consists of all elements which act by zero in all simple left $R$-modules. It follows that the simple modules of $R$ are naturally identified with the simple modules of $R/J(R)$.
The Jacobson radical has an important alternate characterization as consisting of all elements $r$ such that $1 - xr$ is invertible for all $x$. Using this characterization, I claim that the Jacobson radical of $R = U_n(F)$ consists precisely of the strictly upper triangular matrices. The quotient $R/J(R)$ is isomorphic to $F^n$, where the isomorphism sends an upper triangular matrix to its diagonal, and the simple modules of $F^n$ are identified with the $n$ copies of $F$.
$U_n(F)$ happens to be a quiver algebra of a finite acyclic quiver, and a similar statement is true in this generality: the simple modules of such an algebra are naturally in bijection with the vertices of the corresponding quiver.