On representations of the bimodule DA

Abstract

Let $A$ be a finite-dimensional algebra over an algebraically closed field $k$. A representation of the $A$-$A$ bimodule $DA=\mathrm{Hom}_{k}(A, k)$ is a module over the matrix algebra: \[\overline{A}= \begin{bmatrix} A & 0 \\ DA & A \end{bmatrix}\] We first prove that $\overline{A}$ is representation-finite (and in fact simply connected) whenever $A$ is an iterated tilted algebra of Dynbin type. We then assume that $A$ is a tilted algebra of Dynkin type, and characterise $\overline{A}$ by its Auslander-Reiten quiver.