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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 [[#References|[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]] $\mathbf{A} _ { n }$ ($n \in \mathbf N$), ${\bf D} _ { n }$ ($n \in \mathbf N$, $&gt; 4$), or $\mathbf{E} _ { n }$ ($n = 6,7,8$) and $G$ is an infinite [[Cyclic group|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 [[#References|[a4]]] and [[#References|[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|tilting theory]] offered a convenient way for a solution. Namely, it was observed in [[#References|[a1]]] and [[#References|[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|Tilted algebra]]). Nevertheless, the full classification of all these repetitive algebras eventually obtained in [[#References|[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|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 [[#References|[a6]]] and [[#References|[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 [[#References|[a8]]]. In this case tilted algebras of representation-tame hereditary and canonical algebras replace the tilted algebras of representation-finite hereditary algebras.

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  O. Bretscher,   C. Läser,   C. Riedtmann,   "Selfinjective and simply connected algebras"  ''Manuscripta Math.'' , '''36'''  (1981/82)  pp. 253–307</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  D. Hughes,   J. Waschbüsch,   "Trivial extensions of tilted algebras"  ''Proc. London Math. Soc.'' , '''46'''  (1983)  pp. 347–364</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  C. Riedtmann,   "Algebren, Darstellungen, Überlagerungen und zurück"  ''Comment. Math. Helv.'' , '''55'''  (1980)  pp. 199–224</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Riedtmann,   "Representation-finite selfinjective algebras of class $\mathbf{A} _ { n }$" , ''Representation theory II'' , ''Lecture Notes in Mathematics'' , '''832''' , Springer  (1981)  pp. 449–520</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  C. Riedtmann,   "Configurations of $\mathbf{ZD}_n$"  ''J. Algebra'' , '''82'''  (1983)  pp. 309–327</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  C. Riedtmann,   "Representation-finite self-injective algebras of class ${\bf D} _ { n }$"  ''Compositio Math.'' , '''49'''  (1983)  pp. 231–282</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  B. Roggon,   "Selfinjective and iterated tilted algebras of type ${\bf E} _ { 6 }$, $\mathbf{E}_{7}$, $\mathbf{E} _ { 8 }$" , ''E 95-008 SFB'' , '''343''' , Bielefeld  (1995)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  A. Skowroński,   "Selfinjective algebras of polynomial growth"  ''Math. Ann.'' , '''285'''  (1989)  pp. 177–199</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J. Waschbüsch,   "Symmetrische Algebren vom endlichen Modultyp"  ''J. Reine Angew. Math.'' , '''321'''  (1981)  pp. 78–98</td></tr>
</table>

[[Category:Associative rings and algebras]]