The tensorial description of the Auslander algebras of representation-finite string algebras

The aim of this article is to study the Auslander algebra of any representation-finite string algebra. More precisely, we introduce the notion of gluing algebras and show that the Auslander algebra of a representation-finite string algebra is a quotient of a \gluing algebra of ${\vec{A}}_n^e$. As applications, the Auslander algebras of two classes of string algebras whose quivers are Dynkin types $A$ and $D$ are described. Moreover, the representation types of the above Auslander algebras are also given exactly.