Riedtmann classification

A finite-dimensional algebra $A$ over an algebraically closed field $k$ is called self-injective if $A$, considered as a right $A$-module, is injective (cf. also Injective module). Well-known examples for self-injective algebras are the group algebras $k G$ for finite groups $G$ (cf. also Group algebra). An arbitrary finite-dimensional algebra $A$ is said to be representation-finite provided that there are only finitely many isomorphism classes of indecomposable finite-dimensional right $A$-modules.

C. Riedtmann made the main contribution to the classification of all self-injective algebras that are representation-finite. Her key idea was not to look at the algebra $A$ itself, but rather at its Auslander–Reiten quiver $\Gamma_{A}$. (Quiver is an abbreviation for directed graph, see Quiver.) The vertices of the Auslander–Reiten quiver (see also Representation of an associative algebra) are the isomorphism classes of finite-dimensional $A$-modules. The number of arrows from the isomorphism class of $X$ to the isomorphism class of $Y$ is the dimension of the space $\operatorname{rad}_{A}(X, Y) / \operatorname{rad}_{ A } ^ { 2 } ( X , Y )$, where $\operatorname { rad }$ is the Jacobson radical of the category of all finite-dimensional $A$-modules. The Auslander–Reiten quiver is a translation quiver, which means that it carries an additional structure, namely a translation $\tau_{A}$ mapping the non-projective vertices bijectively to the non-injective vertices. The translation is induced by the existence of almost-spit sequences $0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0$ (see also Representation of an associative algebra; Almost-split sequence) and sends the isomorphism class of a non-projective indecomposable module $Z$ to the starting term $X$.

The stable part $( \Gamma _ { A } ) _ { s }$ of the Auslander–Reiten quiver $\Gamma _ { A }$ of $A$ is the full subquiver of $\Gamma _ { A }$ given by the modules that cannot be shifted into an injective or projective vertex by a power $\tau _ { A } ^ { j }$ for some integer $j$. In [a3], Riedtmann succeeded to prove that for any connected representation-finite finite-dimensional $A$ the stable part $( \Gamma _ { A } ) _ { s }$ of the Auslander–Reiten quiver is of the shape $\mathbf{Z} \overset{\rightharpoonup}{ \Delta } / G$, where $\overset{\rightharpoonup} { \Delta }$ is a quiver whose underlying graph $\Delta$ is a Dynkin diagram $\mathbf{A} _ { n }$ ($n \in \mathbf N$), ${\bf D} _ { n }$ ($n \in \mathbf N$, $> 4$), or $\mathbf{E} _ { n }$ ($n = 6,7,8$) and $G$ is an infinite cyclic group of automorphisms of the translation quiver $\mathbf{Z} \overset{\rightharpoonup}{ \Delta }$. The vertices of $\mathbf{Z} \overset{\rightharpoonup}{ \Delta }$ are the pairs $( i , x )$ such that $i$ is an integer and $x$ a vertex of $\overset{\rightharpoonup} { \Delta }$. From $( i , x )$ to $( i , y )$ there are the arrows $( i , \alpha )$ with $\alpha : x \rightarrow y$ an arrow of $\overset{\rightharpoonup} { \Delta }$. In addition, from $( i + 1 , x )$ to $( i , y )$ there exist the arrows $( i , \alpha ) ^ { \prime }$ with $\alpha : y \rightarrow x$ an arrow of $\overset{\rightharpoonup} { \Delta }$. The translation maps $( i , x )$ to $( i + 1 , x )$.

For a self-injective algebra $A$, the only vertices of the Auslander–Reiten quiver that do not belong to $( \Gamma _ { A } ) _ { s }$ are the isomorphism classes of the indecomposable projective (and injective) modules. Thus, one can reconstruct $\Gamma _ { A }$ from $( \Gamma _ { A } ) _ { s }$ by finding in $( \Gamma _ { A } ) _ { s }$ the starting points of arrows of $\Gamma _ { A }$ ending in projective vertices. These combinatorial data are called a configuration. This shows that for finding all possible Auslander–Reiten quivers $\Gamma _ { A }$ of all connected representation-finite self-injective algebras $A$ one has to classify the infinite cyclic automorphism groups $G$ of $\mathbf{Z} \overset{\rightharpoonup}{ \Delta }$ and the $G$-invariant configurations of $\mathbf{Z} \overset{\rightharpoonup}{ \Delta }$ for all Dynkin diagrams. For the Dynkin diagrams $\mathbf{A} _ { n }$ and ${\bf D} _ { n }$ this classification was carried out in [a4] and [a5].

The classification of the possible configurations for the exceptional Dynkin diagrams ${\bf E} _ { 6 }$, $\mathbf{E} _ { 7 }$, $\mathbf{E} _ { 8 }$ turned out to be more difficult. Fortunately, the development of tilting theory offered a convenient way for a solution. Namely, it was observed in [a1] and [a2] that in order to equip $\mathbf{Z} \overset{\rightharpoonup}{ \Delta }$ with all possible configurations, one has to form the Auslander–Reiten quivers of the repetitive algebras of the tilted algebras of representation-finite hereditary algebras of type $\Delta$ (cf. also Tilted algebra). Nevertheless, the full classification of all these repetitive algebras eventually obtained in [a7] required the use of a computer for handling the huge amount of structures appearing in the case $\mathbf{E} _ { 8 }$.

If one finally wants to return from the Auslander–Reiten quiver $\Gamma _ { A }$ to the algebra $A$ itself, one considers the factor of the free $k$-linear category of $\Gamma _ { A }$ by the mesh relations induced by the almost-split sequences. This factor is called the mesh category of $\Gamma _ { A }$. Forming the endomorphism algebra of the direct sum of all projective objects in this mesh category yields $A$ (up to Morita equivalence), provided that $A$ is standard (i.e. the mesh category is equivalent to the category of indecomposable finite-dimensional $A$-modules). Non-standard algebras appear only if the characteristic of the field $k$ is $2$ and $\Delta$ is of type ${\bf D} _ { n }$. They were classified in [a6] and [a9].

It is worth noting that the approach using repetitive algebras was generalized in order to classify the representation-tame self-injective standard algebras of polynomial growth in [a8]. In this case tilted algebras of representation-tame hereditary and canonical algebras replace the tilted algebras of representation-finite hereditary algebras.

References[edit]