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## Representation Theory of Artin Algebras (1994)

Venue: | Studies in Advanced Mathematics |

Citations: | 641 - 10 self |

### Citations

180 |
Infinite root systems, representations of graphs and invariant theory
- Kac
- 1980
(Show Context)
Citation Context ...s with relations [44]. Another important application of using invariant theory is a result of Kac, generalizing results previously known for hereditary algebras of finite and tame representation type =-=[92,93]-=-. We do not want to recall the definitions necessary to state the precise result here, but we just mention that he gives a description of the dimension vectors which occur for indecomposable represent... |

173 |
Unzerlegbare Darstellungen
- Gabriel
(Show Context)
Citation Context ...many of the central theorems along the way, and try to indicate ideas of proofs in some cases. 2.1. Quivers A quiver F = (Fo, I \ ) is a set of vertices Fo and a set of arrows Tl between the vertices =-=[65]-=-. Assume for simplicity that Fo and Tl are finite. For example, O 4-2 212 IDUN REITEN is a quiver. A (finite dimensional) representation of F over A: is a set of finite dimensional ^-vector spaces V(i... |

149 |
Indecomposable representations of graphs and algebras
- Dlab, Ringel
- 1976
(Show Context)
Citation Context ...such that the sequence V(3)-^t V(\)U V(2) -• V'{3) -> 0 is exact.The maps f'a and fp are the natural ones. The above methods have also been useful for the study of quivers of tame representation type =-=[59]-=-, where a classification is in terms of extended Dynkin diagrams as first proved by Nazarova [107] and Donovan and Freislich [62]. 214 IDUN REITEN Since the study of quivers with no oriented cycles an... |

134 |
Tilted algebras
- Happel, Ringel
(Show Context)
Citation Context ...re has also been further work in this direction. For example, for algebras of finite type given by commutative quivers or trees with zero relations, there are quadratic forms with similar properties (=-=[82]-=-, see also [44]). This connection with quadratic forms can be used to show that for these algebras the multiplicity of each simple module as a composition factor is at most 6, by using a result of Ovs... |

111 |
Preprojective modules over Artin algebras
- AUSLANDER, SMALO
- 1980
(Show Context)
Citation Context ...d A' have the same number of nonprojective simple modules. An example of an interesting class of modules defined by almost split sequences is the class of preprojective modules of Auslander and Smalo =-=[27]-=-. Here they generalize some work of Roiter [131] (see [65]). Projective modules have the property that they cover all modules in the sense that there is a surjective map to any module, and the additiv... |

102 | Coxeter functors and Gabriel’s theorem
- Berstein, Gelfand, et al.
- 1973
(Show Context)
Citation Context ...reas of mathematics through the appearance of Dynkin diagrams. This inspired Bernstein, Gelfand and Ponomarev to give an elegant proof of Gabriel's theorem, using the Weyl group associated with | F | =-=[39]-=-. They also establish the one-one correspondence between indecomposable representations and positive roots directly, in the case of finite type. Corresponding to the reflections and Coxeter transforma... |

97 |
Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors
- Brenner, Butler
- 1979
(Show Context)
Citation Context ...e proof of Bernstein, Gelfand and Ponomarev, inspired the module theoretic definition of partial Coxeter functors [18], which has ended up with the nice theory of tilted algebras of Happel and Ringel =-=[49,102,82,42]-=-. The tilted algebras were for example used in the study of selfinjective algebras [138,87,88]. (See [51] for related work.) As for the Coxeter functor, it is closely related to D Tr [48, 68], which p... |

95 |
Auslander-Reiten sequences and representation-finite algebras
- Gabriel
- 1980
(Show Context)
Citation Context ...rings which are not algebras over an algebraically closed field. We refer to the extensive list of references of papers in the period 1968-78 in the Proceedings of ICRA II at Ottawa (the reference of =-=[68]-=-), to the paper based on my lectures at ICRA HI in Puebla 1980 [116], and also to the surveys [13, 68,123,124,145]. Some of the early parts of the development have been treated in a textbook [112]. I ... |

83 |
Almost split sequences in subcategories
- Auslander, Smalø
(Show Context)
Citation Context ...it sequences in other contexts. G. CLASSICAL ORDERS AND COMMUTATIVE RINGS Whereas there are existence theorems for almost split sequences for certain subcategories of mod A when A is an artin algebra =-=[28]-=-, the primary example where an existence theorem has been proved in another area is the lattices over orders over a complete discrete valuation ring R. Denoting by K the quotient field of R and by A a... |

82 |
The universal cover of a representation-finite algebra
- Gabriel
- 1981
(Show Context)
Citation Context ... natural way, there is a functor k(TA) -> ind A. (See Section 3.1 for the topological definition.) A general theory of 'coverings' of AR-quivers was developed by Bongartz and Gabriel [46] and Gabriel =-=[69]-=-. The relationship is studied between /t-algebras A and A' (and more general /c-categories which come up naturally when studying infinite coverings) when FA< is a covering of FA. In particular, if F i... |

79 |
Coherent functors
- Auslander
- 1966
(Show Context)
Citation Context ...egory mod A is not an abelian category, whereas the functor category mod (mod A) is. The study of uniserial functors has been useful for investigating the invariants a and /? discussed in Section 2.3 =-=[25]-=-. For more discussion of the use of functors in representation theory, we refer to the survey of Auslander [11]. 2.5. Coverings It has turned out to be important to study coverings and universal cover... |

63 |
Coxeter functors without diagrams
- Auslander, Platzeck, et al.
- 1979
(Show Context)
Citation Context ...ralizations. In particular, the important role played by the reflection functors in the proof of Bernstein, Gelfand and Ponomarev, inspired the module theoretic definition of partial Coxeter functors =-=[18]-=-, which has ended up with the nice theory of tilted algebras of Happel and Ringel [49,102,82,42]. The tilted algebras were for example used in the study of selfinjective algebras [138,87,88]. (See [51... |

60 |
Algebras and quadratic forms
- Bongartz
- 1983
(Show Context)
Citation Context ...n further work in this direction. For example, for algebras of finite type given by commutative quivers or trees with zero relations, there are quadratic forms with similar properties ([82], see also =-=[44]-=-). This connection with quadratic forms can be used to show that for these algebras the multiplicity of each simple module as a composition factor is at most 6, by using a result of Ovsienko on quadra... |

60 |
Closures of conjugacy classes of matrices are normal, Invent
- Kraft, Procesi
- 1979
(Show Context)
Citation Context ...rve to illustrate matters from REPRESENTATION THEORY OF ARTIN ALGEBRAS 225 an invariant theoretic point of view, since often enough information is known in order to make computations (see for example =-=[1,80,98,125]-=-). B. TOPOLOGY Topological methods have played an important role in the representation theory of algebras in the last few years, through the theory of coverings as discussed in Section 2.5. Since the ... |

58 |
Isolated singularities and almost split sequences
- Auslander
- 1986
(Show Context)
Citation Context ...es are the finitely generated reflexive /^-modules. At the same time establishing an interesting connection between almost split sequences and singularities, Auslander has proved the following result =-=[15]-=-. THEOREM 15. If R is as above, the AR-quiver for the finitely generated reflexive R-modules is an extended Dynkin diagram, if • ^ » • is replaced by • • , and is a Dynkin diagram if the projective R-... |

53 |
Finite representation type is open
- Gabriel
(Show Context)
Citation Context ...eration, over the complex numbers C. C x C x C c y\ xy) 224 IDUN REITEN In this case there are two irreducible components of algebra structures, and there are results of this nature up to dimension 5 =-=[66,105,77]-=-. Some results along these lines were already obtained in the last century (see [79]). Another natural type of question is whether certain classes of algebras or modules correspond to an open or close... |

50 |
Representation theory of finite graphs and associated algebras Carleton Math. Lecture Notes
- Donovan, Freislich
- 1973
(Show Context)
Citation Context ... also been useful for the study of quivers of tame representation type [59], where a classification is in terms of extended Dynkin diagrams as first proved by Nazarova [107] and Donovan and Freislich =-=[62]-=-. 214 IDUN REITEN Since the study of quivers with no oriented cycles and their representations was successful for investigating hereditary algebras and algebras A with r2 = 0, there were efforts to st... |

50 | Indecomposable representations of the Lorentz group - Gelfand, Ponomarev - 1968 |

44 |
Tame and wild matrix problems
- Drozd
(Show Context)
Citation Context ...algebras of infinite type. Special attention has been given to the so-called tame algebras, where it is possible to classify all indecomposable modules. We do not give the formal definition here (see =-=[63]-=-), and we are also leaving out most of the work on tame algebras from this survey. In contrast, an algebra A is called wild if the classification of the indecomposables in mod A essentially contains t... |

38 |
Vinberg’s characterization of Dynkin diagrams using subadditive functions with applications to D Tr-periodic modules
- Happel, Preiser, et al.
- 1979
(Show Context)
Citation Context ...eneralized by Todorov to selfinjective algebras of finite type which are not necessarily algebras over an algebraically closed field [139]. There are also generalizations to algebras of infinite type =-=[139,81]-=-. In her approach, Todorov used elementary length arguments on short exact sequences. IfO -+ A-* BUP-* C -> 0 is an almost split sequence with P projective (possibly zero) and B having no nonzero proj... |

36 |
Trivial extensions of tilted algebras
- Hughes, Waschbüsch
- 1983
(Show Context)
Citation Context ...Coxeter functors [18], which has ended up with the nice theory of tilted algebras of Happel and Ringel [49,102,82,42]. The tilted algebras were for example used in the study of selfinjective algebras =-=[138,87,88]-=-. (See [51] for related work.) As for the Coxeter functor, it is closely related to D Tr [48, 68], which plays an important role because of its connection with almost split sequences. But here the two... |

27 |
Tilted algebras
- Bongartz
- 1980
(Show Context)
Citation Context ...e proof of Bernstein, Gelfand and Ponomarev, inspired the module theoretic definition of partial Coxeter functors [18], which has ended up with the nice theory of tilted algebras of Happel and Ringel =-=[49,102,82,42]-=-. The tilted algebras were for example used in the study of selfinjective algebras [138,87,88]. (See [51] for related work.) As for the Coxeter functor, it is closely related to D Tr [48, 68], which p... |

26 |
The algebraic and geometric classification of associative algebras of dimension five
- Mazzola
- 1979
(Show Context)
Citation Context ...eration, over the complex numbers C. C x C x C c y\ xy) 224 IDUN REITEN In this case there are two irreducible components of algebra structures, and there are results of this nature up to dimension 5 =-=[66,105,77]-=-. Some results along these lines were already obtained in the last century (see [79]). Another natural type of question is whether certain classes of algebras or modules correspond to an open or close... |

23 |
Large modules over artin algebras, in Algebra, topology and categories
- Auslander
- 1976
(Show Context)
Citation Context ...Auslander, has had some very striking consequences in representation theory. It was used in Auslander's generalization of Brauer-Thrall I to left artin rings, and in the proof of the following result =-=[6,7]-=- (see also [127] for (b)). THEOREM 8. (a) If A is an artin algebra of infinite representation type, there exist indecomposable modules which are not finitely generated. (b) If A is a left artin ring o... |

21 |
Modules determined by their composition factors
- Auslander, Reiten
- 1985
(Show Context)
Citation Context ...using a result of Ovsienko on quadratic forms [111] (see also [43]). We also point out that there are more general cases where the indecomposables are uniquely determined by their composition factors =-=[30, 82, 78, 26]-=-. REPRESENTATION THEORY OF ARTIN ALGEBRAS 215 2.2. Matrices The problem of classifying the representations of a given quiver has, by considering fixed bases for the vector spaces, an equivalent formul... |

21 |
Indecomposable representations at characteristic p
- Higman
- 1954
(Show Context)
Citation Context ...s, like the same global dimension. And if the characteristic p of k divides the order of a finite group G, Higman proved that kG is of finite type if and only if the Sylow/^-subgroups of G are cyclic =-=[86]-=-. It is of interest not only to describe the indecomposable modules in mod A, but also the maps between modules. The Brauer-Thrall conjectures, which were first formulated for group algebras, have bee... |

20 |
Functors and morphisms determined by objects
- Auslander
- 1978
(Show Context)
Citation Context ...n A is a ring which is a finitely generated /^-module, such that KA = A. And a A-lattice is a A-module which is a finitely generated free /^-module. The following basic existence theorem was given in =-=[8,9]-=- (and in [129]). THEOREM 14. Let K be afield and A an order in the semisimple K-algebra A. If C is an indecomposable nonprojective A-lattice or A is an indecomposable noninjective A-lattice, there is ... |

20 |
Problems of linear algebra and classification of quadruples of subspaces in a finite-dimentioal vector space
- Gelfand, Ponomarev
- 1970
(Show Context)
Citation Context ... group, Bernstein, Gelfand and Ponomarev define reflection functors and Coxeter functors between categories of representations of quivers. (A special case of Coxeter functors has already been used in =-=[73]-=-.) We give the definition of reflection functor on an easy example, indicating the connection with reflections in the Weyl group. 1 . . 2 1 ' . . 2 ' Let F be the quiver N . 3 sf , and F the quiver ^ ... |

19 | Group representations without groups - Gabriel, Riedtmann - 1979 |

19 |
Geometric methods in representation theory
- Kraft
- 1982
(Show Context)
Citation Context ...n discussing how objects in the representation theory of artin algebras can be interpreted within other fields, we start with the connections with algebraic geometry and invariant theory. We refer to =-=[97]-=- for a thorough treatment. Let Wbe a vector space of finite dimension n over k. Denote by d\gw the associative unitary fc-algebra structures on W. Then alg^ can be considered as a subset of the vector... |

17 |
Critical simply connected algebras
- BONGARTZ
- 1984
(Show Context)
Citation Context ...ing whether an algebra is of finite type for a certain class of algebras containing the standard algebras of finite type. One has to look at a 'universal cover' for A, and a list of algebras given in =-=[44,83,45]-=-. A is proved to be of finite type by checking that the algebras from the list cannot occur as certain types of subcategories. For arbitrary simply connected algebras a graph is associated with the AR... |

17 |
Algebraische reduktion der scharen bilinearer formen
- Kronecker
(Show Context)
Citation Context ...tions of the quiver • »• corresponds to classifying pairs of (n, m)-matrices (A,B), up to simultaneous row and column transformations. This problem was solved by Kronecker already in the last century =-=[99]-=-. Matrix problems can more generally be thought of as classifying sets of matrices up to an equivalence defined by prescribed allowed operations, like elementary row and column transformations. In fac... |

16 |
Algebraic bundles over Pn and problems of linear algebra', Funktsional. Anal, i Prilozhen
- BERNSTEIN, GELFAND, et al.
- 1978
(Show Context)
Citation Context ...ture we mention that there is an intimate relationship between the category of vector bundles over the projective space Pn and the category of graded modules over the exterior algebra onn+1 variables =-=[41]-=-. 3.3. Use of methods Some of the methods developed to deal with problems in the representation theory of artin algebras are useful also outside this area. On one hand problems from other areas may be... |

16 |
The standard form of a representation-finite algebra
- Bretscher, Gabriel
- 1983
(Show Context)
Citation Context ... standard algebras. The first example of a nonstandard algebra was found by Riedtmann [120]. With each ^-algebra A of finite type there is an associated standard algebra A with the same AR-quiver. In =-=[50]-=- Bretscher and Gabriel show how to construct A directly from A. They also give a method for constructing directly a 'universal cover' for A, without considering FA. A fc-algebra A of finite type is sa... |

16 |
Selfinjective and simply connected algebras
- Bretscher, Läser, et al.
- 1981
(Show Context)
Citation Context ...18], which has ended up with the nice theory of tilted algebras of Happel and Ringel [49,102,82,42]. The tilted algebras were for example used in the study of selfinjective algebras [138,87,88]. (See =-=[51]-=- for related work.) As for the Coxeter functor, it is closely related to D Tr [48, 68], which plays an important role because of its connection with almost split sequences. But here the two theories w... |

15 |
Integral representations and diagrams
- Green, Reiner
- 1978
(Show Context)
Citation Context ... of lattices over orders, where for a certain class of orders, called Backstrom orders in [126], the classification problem can be reduced to the classification problem for representations of quivers =-=[76,126]-=-. Also several problems for integral group rings have been reduced to problems about posets [57]. H. DIFFERENTIAL EQUATIONS The classification of the indecomposable representations of the quiver • £• ... |

13 |
The u-algebra of a restricted Lie algebra is Frobenius
- Berkson
- 1964
(Show Context)
Citation Context ...tion of restricted universal enveloping algebra u(L), which is a factor of the ordinary enveloping algebra. These u(L) are artin algebras. Like group algebras these are also known to be selfinjective =-=[38]-=-, and it is known exactly when they are symmetric [134]. The problem when u(L) is of finite type has been studied in [113], where some sufficient conditions for infinite type are given. We refer to [8... |

13 |
Indecomposable modules for finite groups
- Janusz
- 1969
(Show Context)
Citation Context ...eady mentioned that a characterization of when kG is of finite type was given in [86] and a description of the indecomposable modules for group algebras of finite type was given by Janusz and Kupisch =-=[91,100]-=-. For the opposite direction, the recent developments in the representation theory of artin algebras, in particular the theory of almost split sequences, have been used in group representation theory.... |

12 |
A functorial approach to representation theory, Representations of algebras
- Auslander
- 1980
(Show Context)
Citation Context ...lts on almost split sequences, irreducible maps and preprojective modules are nicely formulated and proved in terms of radical and REPRESENTATION THEORY OF ARTIN ALGEBRAS 221 socle series of functors =-=[27,11]-=-. Also, in the study of stable equivalence functorial methods have been used [19, 20]. A main reason here is that the stable category mod A is not an abelian category, whereas the functor category mod... |

11 |
A survey of existence theorems for almost split sequences
- AUSLANDER
- 1986
(Show Context)
Citation Context ...n A is a ring which is a finitely generated /^-module, such that KA = A. And a A-lattice is a A-module which is a finitely generated free /^-module. The following basic existence theorem was given in =-=[8,9]-=- (and in [129]). THEOREM 14. Let K be afield and A an order in the semisimple K-algebra A. If C is an indecomposable nonprojective A-lattice or A is an indecomposable noninjective A-lattice, there is ... |

10 |
Blocks whose defect groups are Klein four groups: a correction
- Erdmann
- 1982
(Show Context)
Citation Context ...o been used to develop a more general theory of group characters [37] and have suggested new problems in group representation theory, like how almost split sequences behave under Green correspondence =-=[12,37,64,115,140]-=-. D. LIE ALGEBRAS In the theory of complex semisimple Lie algebras there is an interesting occurrence of artin algebras. A certain subcategory of representations, containing the Verma modules, the so-... |

10 |
Representation theory of graded Artin algebras
- Gordon, Green
- 1982
(Show Context)
Citation Context ...some A'. The special case of coverings obtained by the action of a group, called Galois coverings, plays an important role [69]. There are here connections with graded modules and skew group algebras =-=[74,75,117]-=-. The ^-algebras A of finite representation type such that ind A is equivalent to k(VA) are particularly nice. Such algebras are up to Morita equivalence uniquely determined by their AR-quiver (with T... |

10 | On categories of indecomposable modules I - HARADA, SAI - 1970 |

10 |
Partially ordered sets of finite type
- KLEINER
- 1972
(Show Context)
Citation Context ...between two such representations are fc-linear maps/: V-*• V such that/(F(jc)) <= V\x) for all JC in /. The following description of when a poset is of finite representation type was given by Kleiner =-=[94]-=-, using work of Nazarova and Roiter [108]. THEOREM 2. Let I be a finite poset. There is only a finite number of indecomposable representations of I over k if and only if I does not contain any of the ... |

9 |
The Green ring of a finite group
- Benson, Parker
- 1984
(Show Context)
Citation Context ...rphic functors from mod A to abelian groups then A ~ B. Closely related to Theorem 5 is the result that if dimfc Hom^X, A) = dim^ HomA(X,B) for all X in ind A then A is isomorphic to B [14] (see also =-=[37]-=-). An important example of the use of almost split sequences in proving classification theorems is the first step in Riedtmann's classification of selfinjective /c-algebras of finite representation ty... |

9 |
The equivalence of certain functors occurring in the representation theory of Artin algebras and species
- Brenner, Butler
- 1976
(Show Context)
Citation Context ...el [49,102,82,42]. The tilted algebras were for example used in the study of selfinjective algebras [138,87,88]. (See [51] for related work.) As for the Coxeter functor, it is closely related to D Tr =-=[48, 68]-=-, which plays an important role because of its connection with almost split sequences. But here the two theories were developed independently, and later turned out to be related (see [116]). The techn... |

8 |
On simply connected algebras
- BAUTISTA, LARRION, et al.
- 1983
(Show Context)
Citation Context ... occur [130]. For those of tree type An there are close connections with generalized tilted algebras of type An [4]. We also mention that an algebraic characterization of simply connected is given in =-=[35]-=-. 2.6. Relationship between the methods We have already seen that there are relationships between some of the methods discussed, for example between matrix problems and representations of quivers. Alm... |

7 |
Almost split sequences whose middle term has at most two indecomposable summands’, Canad
- Auslander, Bautista, et al.
- 1978
(Show Context)
Citation Context ...or algebras of REPRESENTATION THEORY OF ARTIN ALGEBRAS 219 finite representation type, and we mention the following result of Bautista and Brenner as an example [32]. Further results are contained in =-=[121,16,33,56]-=-. THEOREM 6. I/O ->A-+B->C->0isan almost split sequence over an art in algebra of finite type then B has at most four indecomposable summands. If there are four summands then one indecomposable summan... |

7 |
Treue einfach zusammenhangende Algebren I
- Bongartz
- 1982
(Show Context)
Citation Context ...adratic forms can be used to show that for these algebras the multiplicity of each simple module as a composition factor is at most 6, by using a result of Ovsienko on quadratic forms [111] (see also =-=[43]-=-). We also point out that there are more general cases where the indecomposables are uniquely determined by their composition factors [30, 82, 78, 26]. REPRESENTATION THEORY OF ARTIN ALGEBRAS 215 2.2.... |

7 |
Real subspaces of a quaternion vector space
- Dlab, Ringel
- 1978
(Show Context)
Citation Context ...esentations St, i = \,...,n, where V(i) = k and V(j) = 0 f° r j ^ i- This gives an example of how problems of linear algebra can be formulated and studied in terms of representations of quivers. (See =-=[60,61]-=- for other applications to linear algebra.) The category of representations of partially ordered sets also has a nice interpretation as a subcategory of a module category, so that questions about them... |

6 |
Representation theory of artin algebras V. Methods for computing almost split sequences and irreducible morphisms', Comm. Algebra 5
- AUSLANDER, REITEN
- 1977
(Show Context)
Citation Context ...ule theoretic techniques. In addition to the classical module theory available, including homological methods, there is the theory of almost split sequences, as developed in joint work with Auslander =-=[20,21,22,23,24]-=-. We give the basic definition and existence theorem, and indicate some examples of the types of results which have been obtained using almost split sequences. We refer to [116] for a further discussi... |

6 |
The representations of the differential graded categories', Matrix problems
- KLEINER, ROITER
- 1977
(Show Context)
Citation Context ... only those which can be formulated in terms of representations of quivers and partially ordered sets, was done by Kleiner and Roiter, in introducing the concept of differential graded category (DGC) =-=[96]-=-. In [63] Drozd published a proof of the result that every fc-algebra is tame or wild, and here he used the theory of DGC. We also refer to [63] for the definition of tame and wild. 2.3. Almost split ... |

6 |
Unzerlegbare Moduln endlicher Gruppen mit zyklischer p{Sylowgruppe
- Kupisch
- 1969
(Show Context)
Citation Context ...eady mentioned that a characterization of when kG is of finite type was given in [86] and a description of the indecomposable modules for group algebras of finite type was given by Janusz and Kupisch =-=[91,100]-=-. For the opposite direction, the recent developments in the representation theory of artin algebras, in particular the theory of almost split sequences, have been used in group representation theory.... |

5 |
Representation theory of finite dimensional algebras
- Auslander
- 1982
(Show Context)
Citation Context ... references of papers in the period 1968-78 in the Proceedings of ICRA II at Ottawa (the reference of [68]), to the paper based on my lectures at ICRA HI in Puebla 1980 [116], and also to the surveys =-=[13, 68,123,124,145]-=-. Some of the early parts of the development have been treated in a textbook [112]. I am very grateful to Maurice Auslander, Peter Gabriel and Sverre Smalo for constructive criticism and helpful sugge... |

5 |
On the number of terms in the middle of an almost split sequence
- Bautista, Brenner
- 1981
(Show Context)
Citation Context ...s describing the form of an AR-quiver for algebras of REPRESENTATION THEORY OF ARTIN ALGEBRAS 219 finite representation type, and we mention the following result of Bautista and Brenner as an example =-=[32]-=-. Further results are contained in [121,16,33,56]. THEOREM 6. I/O ->A-+B->C->0isan almost split sequence over an art in algebra of finite type then B has at most four indecomposable summands. If there... |

5 |
Représentations indécomposables des ensembles ordonnés, Séminaire Dubreil (Algèbre), 26e année
- Gabriel
(Show Context)
Citation Context ...h block, and the addition of linear combinations of columns from the first block to the columns in the second block. Such matrix problems have the following nice formulation in terms of vector spaces =-=[67]-=-. A representation of a poset I over k is a ̂ -vector space K together with a collection of subspaces V(x) for x in /, such that V{x) £ V(y) whenever x ^y. Maps between two such representations are fc... |

5 |
The theory of matrices (translation from Russian
- GANTMACHER
- 1959
(Show Context)
Citation Context ...osets [57]. H. DIFFERENTIAL EQUATIONS The classification of the indecomposable representations of the quiver • £• (or the corresponding matrix problem) is used in the theory of differential equations =-=[71]-=-. In perturbation theory the infinite dimensional representations of the same quiver are important. When this problem came up in perturbation theory, no previously developed theory existed, and the al... |

5 |
Projective modules in the category
- IRVING
(Show Context)
Citation Context ...h Kt/Ki+l is one of the Vj, and the number of times Vj occurs is equal to the multiplicity of St in Vj. A systematic study of the artin algebras occurring in this way has recently been done by Irving =-=[90]-=-. Also for restricted Lie algebras artin algebras arise. A restricted Lie algebra was introduced by Jacobson, and is a Lie algebra over a field of prime characteristic satisfying some extra conditions... |

4 |
Replication numbers for non-Dynkin sectional subgraphs in finite Auslander-Reiten quivers and some properties of Weyl roots
- BAUTISTA, BRENNER
- 1983
(Show Context)
Citation Context ...or algebras of REPRESENTATION THEORY OF ARTIN ALGEBRAS 219 finite representation type, and we mention the following result of Bautista and Brenner as an example [32]. Further results are contained in =-=[121,16,33,56]-=-. THEOREM 6. I/O ->A-+B->C->0isan almost split sequence over an art in algebra of finite type then B has at most four indecomposable summands. If there are four summands then one indecomposable summan... |

4 |
Grothendieck groups and almost split sequences', Integral representations and applications
- BUTLER
- 1981
(Show Context)
Citation Context ...each exact sequence 0->A->B->C->0, and by K(A) = K(A, 0)/S(A) the Grothendieck group. The following result, due to Butler and Auslander, can be proved using basic properties of almost split sequences =-=[53,14]-=-. THEOREM 5. A w of finite representation type if and only if the A + C—B, where 0->A->B-+C->0isan almost split sequence, generate S(A) or, equivalently, form a basis for S(A). If A and B are in mod A... |

4 |
C.M.: Coherent tubes
- D’Este, Ringel
- 1984
(Show Context)
Citation Context ...or algebras of REPRESENTATION THEORY OF ARTIN ALGEBRAS 219 finite representation type, and we mention the following result of Bautista and Brenner as an example [32]. Further results are contained in =-=[121,16,33,56]-=-. THEOREM 6. I/O ->A-+B->C->0isan almost split sequence over an art in algebra of finite type then B has at most four indecomposable summands. If there are four summands then one indecomposable summan... |

3 |
Quadratic forms on vector spaces
- ARONSZAJN
- 1960
(Show Context)
Citation Context ...nite dimensional representations of the same quiver are important. When this problem came up in perturbation theory, no previously developed theory existed, and the algebraic foundations were laid in =-=[2,3]-=-. Some of these results have been generalized to arbitrary tame hereditary algebras [122]. I. DYNAMICAL SYSTEMS In dynamical systems problems occur which can be formulated in the language of represent... |

3 |
Fixman,Algebraic spectral problems
- Aronszajn, U
- 1968
(Show Context)
Citation Context ...nite dimensional representations of the same quiver are important. When this problem came up in perturbation theory, no previously developed theory existed, and the algebraic foundations were laid in =-=[2,3]-=-. Some of these results have been generalized to arbitrary tame hereditary algebras [122]. I. DYNAMICAL SYSTEMS In dynamical systems problems occur which can be formulated in the language of represent... |

3 | Representation-finite tree algebras - Bongartz, Ringel - 1981 |

3 |
Normal forms of real matrices with respect to complex similarity
- Dlab, Ringel
- 1977
(Show Context)
Citation Context ...esentations St, i = \,...,n, where V(i) = k and V(j) = 0 f° r j ^ i- This gives an example of how problems of linear algebra can be formulated and studied in terms of representations of quivers. (See =-=[60,61]-=- for other applications to linear algebra.) The category of representations of partially ordered sets also has a nice interpretation as a subcategory of a module category, so that questions about them... |

3 |
Group-graded algebras and the zero relation problem
- GREEN
- 1981
(Show Context)
Citation Context ...some A'. The special case of coverings obtained by the action of a group, called Galois coverings, plays an important role [69]. There are here connections with graded modules and skew group algebras =-=[74,75,117]-=-. The ^-algebras A of finite representation type such that ind A is equivalent to k(VA) are particularly nice. Such algebras are up to Morita equivalence uniquely determined by their AR-quiver (with T... |

3 |
Deformations of five-dimensional algebras with unit
- HAPPEL
- 1979
(Show Context)
Citation Context ...eration, over the complex numbers C. C x C x C c y\ xy) 224 IDUN REITEN In this case there are two irreducible components of algebra structures, and there are results of this nature up to dimension 5 =-=[66,105,77]-=-. Some results along these lines were already obtained in the last century (see [79]). Another natural type of question is whether certain classes of algebras or modules correspond to an open or close... |

3 |
Composition factors for indecomposable modules
- HAPPEL
- 1982
(Show Context)
Citation Context ...using a result of Ovsienko on quadratic forms [111] (see also [43]). We also point out that there are more general cases where the indecomposables are uniquely determined by their composition factors =-=[30, 82, 78, 26]-=-. REPRESENTATION THEORY OF ARTIN ALGEBRAS 215 2.2. Matrices The problem of classifying the representations of a given quiver has, by considering fixed bases for the vector spaces, an equivalent formul... |

3 |
Relative invariants and subgeneric orbits of quivers of finite and tame type
- HAPPEL
- 1981
(Show Context)
Citation Context ...rve to illustrate matters from REPRESENTATION THEORY OF ARTIN ALGEBRAS 225 an invariant theoretic point of view, since often enough information is known in order to make computations (see for example =-=[1,80,98,125]-=-). B. TOPOLOGY Topological methods have played an important role in the representation theory of algebras in the last few years, through the theory of coverings as discussed in Section 2.5. Since the ... |

2 |
On the ring of semi-invariants of the representations of an equioriented quiver of type An
- Abeasis
- 1982
(Show Context)
Citation Context ...rve to illustrate matters from REPRESENTATION THEORY OF ARTIN ALGEBRAS 225 an invariant theoretic point of view, since often enough information is known in order to make computations (see for example =-=[1,80,98,125]-=-). B. TOPOLOGY Topological methods have played an important role in the representation theory of algebras in the last few years, through the theory of coverings as discussed in Section 2.5. Since the ... |

2 | Sections in Auslander-Reiten quivers - BAUTISTA - 1980 |

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AND M.MARTINEZ, 'Representations of partially ordered sets and 1-Gorenstein algebras
- BAUTISTA
- 1978
(Show Context)
Citation Context ...ebra.) The category of representations of partially ordered sets also has a nice interpretation as a subcategory of a module category, so that questions about them can be studied within module theory =-=[36]-=-. An attempt to give a definition of a wider class of matrix problems, not only those which can be formulated in terms of representations of quivers and partially ordered sets, was done by Kleiner and... |

2 |
A Category of %modules, Functional Anal
- Bernstein, GeEfand, et al.
- 1976
(Show Context)
Citation Context ...f artin algebras. A certain subcategory of representations, containing the Verma modules, the so-called category 0, is equivalent to the product of categories of the type mod A for an artin algebra A =-=[40]-=-. An interesting property of these algebras is the following. Let Px,...,Pn be the indecomposable projective A-modules and S1,...,Sn the corresponding simple modules. Then there exist indecomposable m... |

2 | On the classification of local integral representations of finite abelian ^-groups - BUTLER - 1975 |

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Representation of Quivers and Moduli of Linear Dynamical Systems
- Hazewinkel
- 1977
(Show Context)
Citation Context ... classification of indecomposables is known. Nevertheless, this point of view gives insight into the nature of the problem in question, and to what extent a reasonable solution is to be expected (see =-=[85,95]-=-). REPRESENTATION THEORY OF ARTIN ALGEBRAS 229 J. HARISH-CHANDRA MODULES Matrix problems occur in the classification of so-called Harish-Chandra modules. For example for SL (2, U) the matrix problem c... |

2 |
Trivial extensions of tilted algebras
- HOSHINO
- 1982
(Show Context)
Citation Context ...Coxeter functors [18], which has ended up with the nice theory of tilted algebras of Happel and Ringel [49,102,82,42]. The tilted algebras were for example used in the study of selfinjective algebras =-=[138,87,88]-=-. (See [51] for related work.) As for the Coxeter functor, it is closely related to D Tr [48, 68], which plays an important role because of its connection with almost split sequences. But here the two... |

2 |
Representations of a tetrad', Izv
- NAZAROVA
- 1967
(Show Context)
Citation Context ...auer-Thrall II when the characteristic of k is not 2 [31]. The special case when a poset In consists of n incomparable elements is the study of vector spaces with n subspaces, already investigated in =-=[106,73]-=-. Here we have a 21 6 IDUN REITEN close connection with representations of quivers, since the representations of the poset /„ are almost the same as the representations of the quiver r • The only inde... |

1 |
HAPPEL,'Generalized tilted algebras of type An
- ASSEMANDD
- 1982
(Show Context)
Citation Context ...n or D n have been classified [46,54], and Rohnes and Smal0 have shown that any tree can occur [130]. For those of tree type An there are close connections with generalized tilted algebras of type An =-=[4]-=-. We also mention that an algebraic characterization of simply connected is given in [35]. 2.6. Relationship between the methods We have already seen that there are relationships between some of the m... |

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Representation dimension ofartin algebras
- AUSLANDER
- 1971
(Show Context)
Citation Context ...or ( , C) in mod (mod A), but it is interesting to investigate when they hold. For example, it is not hard to see that A is of finite type if and only if ( , C) is of finite length for all C in mod A =-=[5]-=-. This functorial point of view, strongly promoted by Auslander, has had some very striking consequences in representation theory. It was used in Auslander's generalization of Brauer-Thrall I to left ... |

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Representation theory ofartin algebras IV. Invariants given by almost split sequences
- AUSLANDER, REITEN
- 1977
(Show Context)
Citation Context ...Auslander, has had some very striking consequences in representation theory. It was used in Auslander's generalization of Brauer-Thrall I to left artin rings, and in the proof of the following result =-=[6,7]-=- (see also [127] for (b)). THEOREM 8. (a) If A is an artin algebra of infinite representation type, there exist indecomposable modules which are not finitely generated. (b) If A is a left artin ring o... |

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Relations for Grothendieck groups ofartin algebras
- AUSLANDER
- 1984
(Show Context)
Citation Context ...each exact sequence 0->A->B->C->0, and by K(A) = K(A, 0)/S(A) the Grothendieck group. The following result, due to Butler and Auslander, can be proved using basic properties of almost split sequences =-=[53,14]-=-. THEOREM 5. A w of finite representation type if and only if the A + C—B, where 0->A->B-+C->0isan almost split sequence, generate S(A) or, equivalently, form a basis for S(A). If A and B are in mod A... |

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Representation theory of hereditary artin algebras
- AUSLANDER, PLATZECK
- 1976
(Show Context)
Citation Context ...e studied for AR-quivers. Further, the work on representations of quivers served as an inspiration for using almost split sequences to give an alternative approach to the study of hereditary algebras =-=[17]-=-, leading to generalizations. In particular, the important role played by the reflection functors in the proof of Bernstein, Gelfand and Ponomarev, inspired the module theoretic definition of partial ... |

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Stable equivalence ofartin algebras
- AUSLANDER, REITEN
- 1972
(Show Context)
Citation Context ... formulated and proved in terms of radical and REPRESENTATION THEORY OF ARTIN ALGEBRAS 221 socle series of functors [27,11]. Also, in the study of stable equivalence functorial methods have been used =-=[19, 20]-=-. A main reason here is that the stable category mod A is not an abelian category, whereas the functor category mod (mod A) is. The study of uniserial functors has been useful for investigating the in... |

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Stable equivalence of dualizing /^-varieties I
- AUSLANDER, REITEN
- 1974
(Show Context)
Citation Context ...ule theoretic techniques. In addition to the classical module theory available, including homological methods, there is the theory of almost split sequences, as developed in joint work with Auslander =-=[20,21,22,23,24]-=-. We give the basic definition and existence theorem, and indicate some examples of the types of results which have been obtained using almost split sequences. We refer to [116] for a further discussi... |

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Representation theory ofartin algebras VI. A functorial approach to almost split sequences
- AUSLANDER, REITEN
- 1977
(Show Context)
Citation Context ...ule theoretic techniques. In addition to the classical module theory available, including homological methods, there is the theory of almost split sequences, as developed in joint work with Auslander =-=[20,21,22,23,24]-=-. We give the basic definition and existence theorem, and indicate some examples of the types of results which have been obtained using almost split sequences. We refer to [116] for a further discussi... |

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Sections in Auslander-Reiten components IF, An
- BAUTISTA
(Show Context)
Citation Context ...using a result of Ovsienko on quadratic forms [111] (see also [43]). We also point out that there are more general cases where the indecomposables are uniquely determined by their composition factors =-=[30, 82, 78, 26]-=-. REPRESENTATION THEORY OF ARTIN ALGEBRAS 215 2.2. Matrices The problem of classifying the representations of a given quiver has, by considering fixed bases for the vector spaces, an equivalent formul... |

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On algebras of strongly unbounded type
- BAUTISTA
(Show Context)
Citation Context ...able modules [137]. Using this and recent work by Bautista, Gabriel, Roiter and Salmeron (see 2.5), Bautista has recently given another proof of Brauer-Thrall II when the characteristic of k is not 2 =-=[31]-=-. The special case when a poset In consists of n incomparable elements is the study of vector spaces with n subspaces, already investigated in [106,73]. Here we have a 21 6 IDUN REITEN close connectio... |

1 |
Auslander-Reiten quivers for artin algebras of finite representation type
- BAUTISTA, LARRION
- 1982
(Show Context)
Citation Context ...d to be simply connected if FA is its own universal covering [46]. For example all algebras of finite type given by trees with zero relations or by fully commutative quivers are simply connected (see =-=[34,69]-=-). Covering theory has provided an important tool for reducing problems to the simply connected case. As an example of a theorem proved by using this reduction we mention the following result of de la... |

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Simply connected algebras of tree-class An and D n \ preprint
- CONTI, GABRIEL
(Show Context)
Citation Context ...d this graph is a tree [46]. This tree is given a grading, which uniquely determines A (up to Morita equivalence). The simply connected algebras whose trees are of type An or D n have been classified =-=[46,54]-=-, and Rohnes and Smal0 have shown that any tree can occur [130]. For those of tree type An there are close connections with generalized tilted algebras of type An [4]. We also mention that an algebrai... |

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Automorphisms of representation-finite algebras
- PENA, MARTINEZ
- 1983
(Show Context)
Citation Context ...has provided an important tool for reducing problems to the simply connected case. As an example of a theorem proved by using this reduction we mention the following result of de la Pena and Martinez =-=[55]-=-. THEOREM 10. Let A be a k-algebra of finite representation type and g an automorphism of ind A. If g acts freely on all indecomposable projectives P, that is g(P) ^ P, then g acts freely on all indec... |

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Group rings of wild representation type
- DIETRICH
- 1983
(Show Context)
Citation Context ... classification problem can be reduced to the classification problem for representations of quivers [76,126]. Also several problems for integral group rings have been reduced to problems about posets =-=[57]-=-. H. DIFFERENTIAL EQUATIONS The classification of the indecomposable representations of the quiver • £• (or the corresponding matrix problem) is used in the theory of differential equations [71]. In p... |

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Construction of Auslander-Reiten quivers for a class of group rings
- DIETRICH
- 1983
(Show Context)
Citation Context ... when an order is of finite lattice type is central also 228 IDUN REITEN here, and the almost split sequences provide useful criteria, after extending the corresponding algebra theory to this setting =-=[141,142,140,58]-=-. We refer to [128] for a survey. There are also existence theorems for lattices over the higher dimensional orders introduced by Auslander, in particular for certain subcategories of modules over com... |

1 |
Klassifikationstheorie endlich-dimensionaler algebren in der Zeit von 1880 bis 1920', Enseign
- HAPPEL
- 1980
(Show Context)
Citation Context ...are two irreducible components of algebra structures, and there are results of this nature up to dimension 5 [66,105,77]. Some results along these lines were already obtained in the last century (see =-=[79]-=-). Another natural type of question is whether certain classes of algebras or modules correspond to an open or closed subset of the associated affine variety. For example, an orbit CM corresponding to... |

1 |
Minimal algebras of finite representation type with preprojective component
- HAPPEL, VOSSIECK
- 1983
(Show Context)
Citation Context ...ing whether an algebra is of finite type for a certain class of algebras containing the standard algebras of finite type. One has to look at a 'universal cover' for A, and a list of algebras given in =-=[44,83,45]-=-. A is proved to be of finite type by checking that the algebras from the list cannot occur as certain types of subcategories. For arbitrary simply connected algebras a graph is associated with the AR... |

1 |
Restricted Lie algebras (and beyond
- HUMPHREYS
- 1982
(Show Context)
Citation Context ... for restricted Lie algebras artin algebras arise. A restricted Lie algebra was introduced by Jacobson, and is a Lie algebra over a field of prime characteristic satisfying some extra conditions (see =-=[89]-=-). There is also the notion of restricted universal enveloping algebra u(L), which is a factor of the ordinary enveloping algebra. These u(L) are artin algebras. Like group algebras these are also kno... |

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Representations of quivers and canonical forms for linear systems
- KLEINER, MARTIN
- 1981
(Show Context)
Citation Context ... classification of indecomposables is known. Nevertheless, this point of view gives insight into the nature of the problem in question, and to what extent a reasonable solution is to be expected (see =-=[85,95]-=-). REPRESENTATION THEORY OF ARTIN ALGEBRAS 229 J. HARISH-CHANDRA MODULES Matrix problems occur in the classification of so-called Harish-Chandra modules. For example for SL (2, U) the matrix problem c... |

1 |
Representations indecomposables des ensembles ordonnes finis', These, Universite de Tours
- LOUPIAS
- 1975
(Show Context)
Citation Context ...lly commutative when there are no oriented cycles, and whenever two paths have the same start and end point, the corresponding compositions of maps should be the same. This case was solved by Loupias =-=[101]-=- and Skabara and Zavadskij [135]. (2) Algebras given by quivers which are trees. Here the only possible relations are zero relations, and a classification of finite type was given by Bongartz and Ring... |

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Reflection functors
- MARMARIDIS
- 1980
(Show Context)
Citation Context ...e proof of Bernstein, Gelfand and Ponomarev, inspired the module theoretic definition of partial Coxeter functors [18], which has ended up with the nice theory of tilted algebras of Happel and Ringel =-=[49,102,82,42]-=-. The tilted algebras were for example used in the study of selfinjective algebras [138,87,88]. (See [51] for related work.) As for the Coxeter functor, it is closely related to D Tr [48, 68], which p... |

1 | Algebras stably equivalent to factors of hereditary - MARTINEZ - 1981 |

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The stable equivalence for algebras of finite representation type
- MARTINEZ
(Show Context)
Citation Context ... simples in general, and this would be especially interesting to know for group algebras. It is known in some situations, and one general theorem along these lines is the following result of Martinez =-=[104]-=-. THEOREM 7. If A and A' are stably equivalent k-algebras of finite type then A and A' have the same number of nonprojective simple modules. An example of an interesting class of modules defined by al... |