View source for Representation of an associative algebra

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r0814302.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r0814303.png" />''

A homomorphism of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r0814304.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r0814305.png" /> into the algebra of matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r0814306.png" />, i.e. a mapping assigning to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r0814307.png" /> a square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r0814308.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r0814309.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143012.png" />. It is usually also required that the unit in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143013.png" /> corresponds to the identity matrix; sometimes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143014.png" /> is also required to be finite-dimensional.

Every indecomposable representation of a finite-dimensional semi-simple algebra is equivalent to a direct summand of the [[Regular representation|regular representation]]. Hence, every finite-dimensional semi-simple algebra is an algebra of finite (representation) type, i.e. has a finite number of non-isomorphic indecomposable representations. Non-semi-simple algebras can be both of finite and of infinite representation type (e.g. such is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143015.png" />). Algebras of infinite type are further divided into algebras of wild type, whose classification problem contains the unsolved problem on matrix pairs (i.e. the problem of simultaneously reducing to canonical form two linear operators on a finite-dimensional space), and algebras of tame type.

Basic problems studied in the representation theory of associative algebras are that of obtaining necessary and sufficient conditions for an algebra to belong to one of the types listed, as well as that of classifying the indecomposable representations in the finite and tame cases. In the general case these problems have not been solved. The description of algebras of finite or tame type and their representations has been obtained for algebras the square of whose radical equals zero (cf. [[#References|[2]]], [[#References|[4]]], [[#References|[8]]]–[[#References|[10]]]). The Brauer–Thrall problem has been solved, i.e. it has been proved that, over any field, an algebra of infinite type has indecomposable representations of arbitrary high dimension, while over a perfect field there are infinitely many dimensions in each of which there are infinitely many indecomposable representations (cf. [[#References|[5]]], [[#References|[7]]]). Any algebra of finite type over an algebraically closed field has a multiplicative basis, i.e. a basis for which the product of two arbitrary elements in it is either zero or belongs to the basis [[#References|[6]]]. The problem of dividing the class of group algebras into tame and wild ones has been completely solved [[#References|[1]]].

Strongly related with representations of associative algebras are representations of other objects: rings, partially ordered sets, lattices, boxes.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.M. Bondarenko,   Yu.A. Drozd,   "Representation type of finite groups"  ''J. Soviet Math.'' , '''20''' :  6  (1982)  pp. 2515–2528  ''Zap. Nauchn. Sem. Leningr. Univ.'' , '''71'''  (1977)  pp. 24–41</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.A. Kruglyak,   "Representations of algebras the square of whose radical equals zero"  ''J. Soviet Math.'' , '''3''' :  5  (1975)  pp. 629–636  ''Zap. Nauchn. Sem. Leningr. Univ.'' , '''28'''  (1972)  pp. 60–69</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.A. Nazarova,   "Representations of quivers of infinite type"  ''Math. USSR Izv.'' , '''7''' :  4  (1973)  pp. 749–792  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''37''' :  4  (1973)  pp. 752–791</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  U. Fischbacher,   "Une nouvelle preuve d'un théorème de Nazarova et Roiter"  ''C.R. Acad. Sci. Paris'' , '''300'''  (1984)  pp. 259–263</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Bautista,   P. Gabriel,   A. Roiter,   L. Salmeron,   "Representation finite algebras and multiplicative bases"  ''Invent. Math.'' , '''81'''  (1985)  pp. 217–285</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.V. Roiter,   "Unbounded dimensionality of indecomposable representations of an algebra with an infinite number of indecomposable representations"  ''Math. USSR Izv.'' , '''2''' :  6  (1968)  pp. 1223–1230  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''32''' :  6  (1968)  pp. 1275–1282</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V. Dlab,   C. Ringel,   "Indecomposable representations of graphs and algebras" , Amer. Math. Soc.  (1976)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P. Donovan,   M.R. Freislich,   "The representation theory of finite graphs and associated algebras" , Carleton Univ.  (1974)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  P. Gabriel,   "Unzerlegbare Darstellungen I"  ''Manuscripta Math.'' , '''6''' :  1  (1972)  pp. 71–103</TD></TR></table>



====Comments====
Thus, for an associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143016.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143017.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143018.png" />), a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143019.png" /> is an algebra homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143021.png" /> is a vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143023.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143024.png" />-algebra of all (linear) endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143025.png" />. A subrepresentation of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143026.png" /> is given by a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143028.png" /> which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143029.png" />-invariant for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143030.png" />, and, in this case, one obtains a representation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143031.png" />, called a quotient representation. Given a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143033.png" />, there is the dual (or contragredient) representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143034.png" /> of the opposite algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143036.png" /> (this is the algebra on the underlying vector space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143037.png" /> with multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143038.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143039.png" />); by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143040.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143043.png" />. Cf. also [[Contragredient representation|Contragredient representation]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143044.png" /> be a representation; instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143045.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143047.png" />, one often writes just <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143048.png" />; in this way <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143049.png" /> becomes a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143050.png" />-module, and any left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143051.png" />-module is obtained in this way. Given two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143053.png" />, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143054.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143055.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143056.png" /> is a linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143057.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143058.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143060.png" />, or, rewritten, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143061.png" />; thus it is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143062.png" />-module homomorphism. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143063.png" /> is a family of representations, their direct sum is the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143065.png" /> is the direct sum of vector spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143066.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143067.png" />. The category of all representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143068.png" />, equivalently, the category of (left) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143069.png" />-modules, is an Abelian category. Note that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143070.png" /> is a central idempotent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143071.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143073.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143074.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143075.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143076.png" />-module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143078.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143079.png" />-modules, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143080.png" /> is the direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143082.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143083.png" />. On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143084.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143086.png" />, and one may regard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143087.png" /> as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143088.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143089.png" /> as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143090.png" />-module. Thus, dealing with representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143091.png" /> one may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143092.png" /> is connected (that is, the only central idempotents of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143093.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143095.png" />).

A representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143096.png" /> is said to be simple (or irreducible) provided it is non-zero and the only proper subrepresentation is the zero representation. The Schur lemma asserts that the endomorphism ring of a simple representation is a division ring (cf. [[Ring with division|Ring with division]]). A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143097.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143098.png" /> is said to be of finite length if there is a sequence of subrepresentations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r08143099.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430100.png" /> is simple, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430101.png" />; such a sequence is called a composition series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430103.png" /> is its length, and the factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430104.png" /> are called the composition factors (cf. also [[Composition sequence|Composition sequence]]). If a representation has a composition series, then any two composition series have the same length, and there is a bijection between the composition factors of the two series (the [[Jordan–Hölder theorem|Jordan–Hölder theorem]]). This may be formulated also as follows: the [[Grothendieck group|Grothendieck group]] of all finite-length representations modulo exact sequences is the free Abelian group on the set of isomorphism classes of simple representations. A representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430105.png" /> is called semi-simple if it is a direct sum of simple representations, or, equivalently, if any subrepresentation is a direct summand.

A representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430106.png" /> is said to be indecomposable if it cannot be written as the direct sum of two non-zero representations. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430107.png" /> is an indecomposable representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430108.png" /> of finite length, then its endomorphism ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430109.png" /> is a [[Local ring|local ring]]. For a finite direct sum of representations with local endomorphism rings, all direct sum decompositions into indecomposable representations are equivalent (the Krull–Schmidt theorem, cf. [[Krull–Remak–Schmidt theorem|Krull–Remak–Schmidt theorem]]). It follows that the Grothendieck group of all finite-length modules modulo split exact sequences is the free Abelian group on the set of isomorphism classes of indecomposable representations.

The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430110.png" /> is said to be representation-finite if there are only finitely many isomorphism classes of indecomposable representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430111.png" />; it is called tame if it is not representation-finite but all families of indecomposable representations are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430112.png" />-parametric, and wild if the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430113.png" />-mod of all finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430114.png" />-modules involves the classification problem for pairs of square matrices up to simultaneous equivalence [[#References|[a7]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430115.png" /> be a finite-dimensional algebra. If the dimension of each finite-dimensional indecomposable representation is bounded, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430116.png" /> is representation-finite (the first Brauer–Thrall conjecture, solved by V.A. Roiter [[#References|[7]]]) and any representation is the direct sum of finite-dimensional indecomposable ones [[#References|[a12]]]. The second Brauer–Thrall conjecture asserts that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430117.png" /> is not representation-finite and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430118.png" /> is an infinite field, then there are infinitely many isomorphism classes of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430119.png" />, for infinitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430120.png" />. The conjecture has been solved for perfect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430121.png" /> by R. Bautista [[#References|[a3]]] and K. Bongartz [[#References|[a5]]], see also [[#References|[a11]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430122.png" /> is not representation-finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430123.png" /> is either tame or wild and not both (Drozd's theorem [[#References|[a7]]]). Certain minimal representation-infinite algebras have been classified by D. Happel and D. Vossieck [[#References|[a9]]], and questions concerning minimal representation-infinite algebras over algebraically closed fields can be transferred to this list; in particular, one gets in this way a criterion for finite-representation type [[#References|[a4]]], [[#References|[a8]]]. In general, questions concerning finite-dimensional algebras over algebraically closed fields are treated by considering quivers with relations (see [[Quiver|Quiver]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430124.png" /> be a finite-dimensional algebra. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430125.png" /> has no non-zero nilpotent ideal, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430126.png" /> is said to be semi-simple. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430127.png" /> is semi-simple if and only if any representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430128.png" /> is semi-simple; in this case, the simple representations are just the indecomposable summands of the [[Regular representation|regular representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430129.png" />. In general, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430130.png" /> be the radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430131.png" /> (cf. [[Radical of rings and algebras|Radical of rings and algebras]]), it is the maximal nilpotent ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430132.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430133.png" /> is semi-simple. The simple representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430134.png" /> are the indecomposable summands of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430135.png" />; up to isomorphism, there are only finitely many. The indecomposable projective representations are the direct summands of the regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430136.png" />, the indecomposable injective representations are the duals of the regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430137.png" />. Any indecomposable [[Projective representation|projective representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430138.png" /> has a unique simple quotient representation, any indecomposable injective representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430139.png" /> has a unique simple subrepresentation; in this way one obtains a bijection between the isomorphism classes of the simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430140.png" />-modules and the indecomposable projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430141.png" />-modules, as well as the indecomposable injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430142.png" />-modules.

The basic notions of modern representation theory are due to M. Auslander and I. Reiten [[#References|[a1]]]: Given any indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430143.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430144.png" />, there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430145.png" /> which is minimal right almost split: it is not a split epimorphism, given any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430146.png" /> which is not a split endomorphism, there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430147.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430148.png" />, and given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430149.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430150.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430151.png" /> is an automorphism. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430152.png" /> is projective, take for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430153.png" /> its maximal submodule and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430154.png" /> the inclusion mapping. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430155.png" /> non-projective, the minimal right almost split mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430156.png" /> is surjective, its kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430157.png" /> is indecomposable (and not injective), and the inclusion mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430158.png" /> is minimal left almost split (defined by the dual properties); also, any indecomposable non-injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430159.png" />-module occurs in this way as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430160.png" />. These exact sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430161.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430162.png" /> minimal left almost split and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430163.png" /> minimal right almost split are called almost-split sequences (or Auslander–Reiten sequences). They are uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430164.png" /> and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430165.png" />; given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430166.png" />, the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430167.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430168.png" /> can be calculated as follows: Take a minimal projective representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430169.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430170.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430171.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430172.png" />; the construction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430173.png" /> is called the Auslander–Reiten translation.

The Auslander–Reiten quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430174.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430175.png" /> has as vertices the isomorphism classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430176.png" /> of the finite-dimensional indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430177.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430178.png" />, and there is an arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430179.png" /> provided there exists an irreducible mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430180.png" /> (note that a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430181.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430182.png" /> indecomposable is called irreducible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430183.png" /> is not invertible and given a factorization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430184.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430185.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430186.png" /> is a split monomorphism or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430187.png" /> is a split epimorphism); in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430188.png" /> is equipped with the Auslander–Reiten translation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430189.png" />. The meshes of the Auslander–Reiten quiver are as follows: Given an indecomposable non-projective representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430190.png" /> and an indecomposable representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430191.png" />, there is an irreducible mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430192.png" /> if and only if there is an irreducible mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430193.png" /> (this is the case if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430194.png" /> is a direct summand of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430195.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430196.png" /> is the middle term of the almost-split sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430197.png" />). The Auslander–Reiten quiver of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430198.png" /> is an important combinatorial invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430199.png" />, often one may recover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430200.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430201.png" />. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430202.png" /> is connected and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430203.png" /> has a finite component, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430204.png" /> is representation-finite (Auslander's theorem, [[#References|[a1]]]). Deleting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430205.png" /> the vertices of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430206.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430207.png" /> indecomposable injective, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430208.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430209.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430210.png" /> indecomposable projective, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430211.png" />, one obtains the stable Auslander–Reiten quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430212.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430213.png" /> representation-finite, the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430214.png" /> are related to Dynkin diagrams (cf. [[Dynkin diagram|Dynkin diagram]]) [[#References|[a13]]], [[#References|[a10]]]. Using covering theory [[#References|[a6]]], the study of representation-finite algebras can be reduced to that of representation-directed algebras (an algebra is called representation directed if there are only finitely many indecomposable representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430215.png" /> and they can be ordered so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430216.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430217.png" />). The Auslander–Reiten quiver of a representation-directed algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430218.png" /> (and therefore the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430219.png" />-mod) can be constructed effectively [[#References|[a14]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Auslander,   "Applications of morphisms determined by objects"  R. Gordon (ed.) , ''Representation Theory of Algebras'' , M. Dekker  (1978)  pp. 245–327</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Auslander,   I. Reiten,   "Representation theory of Artin algebras III"  ''Comm. in Algebra''  (1975)  pp. 239–294</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Bautista,   "On algebras of strongly unbounded representation type"  ''Comment. Math. Helv.'' , '''60'''  (1985)  pp. 392–399</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Bongartz,   "A criterion for finite representation type"  ''Math. Ann.'' , '''269'''  (1984)  pp. 1–12</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Bongartz,   "Indecomposables are standard"  ''Comment. Math. Helv.'' , '''60'''  (1985)  pp. 400–410</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Bongartz,   P. Gabriel,   "Covering spaces in representation theory"  ''Invent. Math.'' , '''65'''  (1981)  pp. 381–387</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Yu.A. Drozd,   "Tame and wild matrix problems"  V. Dlab (ed.)  P. Gabriel (ed.) , ''Representation Theory II'' , ''Lect. notes in math.'' , '''832''' , Springer  (1980)  pp. 242–258</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  P. Dräxler,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081430/r081430220.png" />-Fasersummen in darstellungsendlichen Algebren"  ''J. Algebra'' , '''113'''  (1988)  pp. 430–437</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D. Happel,   D. Vossieck,   "Minimal algebras of infinite representation type with preprojective component"  ''Manuscripta Math.'' , '''42'''  (1983)  pp. 221–243</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  D. Happel,   U. Preiser,   C.M. Ringel,   "Vinberg's characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules"  V. Dlab (ed.)  P. Gabriel (ed.) , ''Representation Theory II'' , ''Lect. notes in math.'' , '''832''' , Springer  (1980)  pp. 280–294</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  L.A. Nazarova,   A.V. Roiter,   "Categorical matrix problems and the Brauer–Thrall conjecture" , Kiev  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  C.M. Ringel,   H. Tachikawa,   "QF-3 rings"  ''J. Reine Angew. Math.'' , '''272'''  (1975)  pp. 49–72</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  Chr. Riedtmann,   "Algebren, Darstellungsköcher, Überlagerungen, und zurück"  ''Comment. Math. Helv.'' , '''55'''  (1980)  pp. 199–224</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  C.M. Ringel,   "Tame algebras and integral quadratic forms" , ''Lect. notes in math.'' , '''1099''' , Springer  (1984)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  D. Happel,   "Triangulated categories in representation theory of finite dimensional algebras" , London Math. Soc.  (1988)</TD></TR></table>

''C.M. Ringel''