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2018 | OriginalPaper | Chapter

Auslander–Reiten Theory for Finite-Dimensional Algebras

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Abstract

This article is based on a course given at the CIMPA School “Homological Methods, Representation Theory and Cluster Algebras,” held in March 2016 in Mar del Plata. The aim of the course, consisting of four lectures, was to provide a brief introduction to the notion of an almost split sequence and its use in the representation theory of finite-dimensional algebras. The first two sections are reduced, and the next three sections are extended in comparison with the above-mentioned course.

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Literature
2.
go back to reference Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory, London Math. Soc. Stud. Texts, vol. 65. Cambridge Univ. Press, Cambridge (2006). DOI https://doi.org/10.1017/CBO9780511614309 Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory, London Math. Soc. Stud. Texts, vol. 65. Cambridge Univ. Press, Cambridge (2006). DOI https://​doi.​org/​10.​1017/​CBO9780511614309​
6.
go back to reference Assem, I., Skowroński, A.: Multicoil algebras. In: V. Dlab, H. Lenzing (eds.) Representations of Algebras (Ottawa, ON, 1992), CMS Conf. Proc., vol. 14, pp. 29–68. Amer. Math. Soc., Providence, RI (1993) Assem, I., Skowroński, A.: Multicoil algebras. In: V. Dlab, H. Lenzing (eds.) Representations of Algebras (Ottawa, ON, 1992), CMS Conf. Proc., vol. 14, pp. 29–68. Amer. Math. Soc., Providence, RI (1993)
9.
go back to reference Auslander, M., Bridger, M.: Stable module theory. Mem. Amer. Math. Soc. (94) (1969) Auslander, M., Bridger, M.: Stable module theory. Mem. Amer. Math. Soc. (94) (1969)
17.
go back to reference Bautista, R.: Irreducible morphisms and the radical of a category. An. Inst. Mat. Univ. Nac. Autónoma México 22, 83–135 (1982) Bautista, R.: Irreducible morphisms and the radical of a category. An. Inst. Mat. Univ. Nac. Autónoma México 22, 83–135 (1982)
18.
go back to reference Bautista, R., Brenner, S.: On the number of terms in the middle of an almost split sequence. In: M. Auslander, E. Lluis (eds.) Representations of Algebras (Puebla, 1980), Lecture Notes in Math., vol. 903, pp. 1–8. Springer, Berlin (1981). DOI https://doi.org/10.1007/BFb0092980 Bautista, R., Brenner, S.: On the number of terms in the middle of an almost split sequence. In: M. Auslander, E. Lluis (eds.) Representations of Algebras (Puebla, 1980), Lecture Notes in Math., vol. 903, pp. 1–8. Springer, Berlin (1981). DOI https://​doi.​org/​10.​1007/​BFb0092980
20.
go back to reference Bernsten, I.N., Gel’fand, I.M., Ponomarev, V.A.: Coxeter functors, and Gabriel’s theorem (Russian). Uspehi Mat. Nauk 28(2(170)), 19–33 (1973). English transl., Russian Math. Surveys 28(2), 17–32 (1973) Bernsten, I.N., Gel’fand, I.M., Ponomarev, V.A.: Coxeter functors, and Gabriel’s theorem (Russian). Uspehi Mat. Nauk 28(2(170)), 19–33 (1973). English transl., Russian Math. Surveys 28(2), 17–32 (1973)
24.
go back to reference Brenner, S., Butler, M.C.R.: Wild subquivers of the Auslander–Reiten quiver of a tame algebra. In: E.L. Green, B. Huisgen-Zimmermann (eds.) Trends in the Representation Theory of Finite-Dimensional Algebras (Seattle, WA, 1997), Contemp. Math., vol. 229, pp. 29–48. Amer. Math. Soc., Providence, RI (1998). DOI https://doi.org/10.1090/conm/229/03309 Brenner, S., Butler, M.C.R.: Wild subquivers of the Auslander–Reiten quiver of a tame algebra. In: E.L. Green, B. Huisgen-Zimmermann (eds.) Trends in the Representation Theory of Finite-Dimensional Algebras (Seattle, WA, 1997), Contemp. Math., vol. 229, pp. 29–48. Amer. Math. Soc., Providence, RI (1998). DOI https://​doi.​org/​10.​1090/​conm/​229/​03309
25.
go back to reference Bretscher, O., Gabriel, P.: The standard form of a representation-finite algebra. Bull. Soc. Math. France 111(1), 21–40 (1983)MathSciNetCrossRef Bretscher, O., Gabriel, P.: The standard form of a representation-finite algebra. Bull. Soc. Math. France 111(1), 21–40 (1983)MathSciNetCrossRef
30.
go back to reference Crawley-Boevey, W.: Modules of finite length over their endomorphism rings. In: H. Tachikawa, S. Brenner (eds.) Representations of Algebras and Related Topics (Kyoto, 1990), London Math. Soc. Lecture Note Ser., vol. 168, pp. 127–184. Cambridge Univ. Press, Cambridge (1992) Crawley-Boevey, W.: Modules of finite length over their endomorphism rings. In: H. Tachikawa, S. Brenner (eds.) Representations of Algebras and Related Topics (Kyoto, 1990), London Math. Soc. Lecture Note Ser., vol. 168, pp. 127–184. Cambridge Univ. Press, Cambridge (1992)
31.
go back to reference Dlab, V., Gabriel, P. (eds.): Representation Theory. II (Ottawa, ON, 1979), Lecture Notes in Math., vol. 832. Springer, Berlin (1980) Dlab, V., Gabriel, P. (eds.): Representation Theory. II (Ottawa, ON, 1979), Lecture Notes in Math., vol. 832. Springer, Berlin (1980)
34.
go back to reference Dlab, V., Ringel, C.M.: The representations of tame hereditary algebras. In: Gordon [38], pp. 329–353 Dlab, V., Ringel, C.M.: The representations of tame hereditary algebras. In: Gordon [38], pp. 329–353
35.
go back to reference Donovan, P., Freislich, M.R.: The Representation Theory of Finite Graphs and Associated Algebras, Carleton Math. Lecture Notes, vol. 5. Carleton Univ., Ottawa, ON (1973) Donovan, P., Freislich, M.R.: The Representation Theory of Finite Graphs and Associated Algebras, Carleton Math. Lecture Notes, vol. 5. Carleton Univ., Ottawa, ON (1973)
38.
go back to reference Gordon, R. (ed.): Representation Theory of Algebras (Philadelphia, PA, 1976), Lecture Notes in Pure Appl. Math., vol. 37. Dekker, New York (1978) Gordon, R. (ed.): Representation Theory of Algebras (Philadelphia, PA, 1976), Lecture Notes in Pure Appl. Math., vol. 37. Dekker, New York (1978)
39.
go back to reference Happel, D., Preiser, U., Ringel, C.M.: Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to \(D\) Tr-periodic modules. In: Dlab and Gabriel [31], pp. 280–294 Happel, D., Preiser, U., Ringel, C.M.: Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to \(D\) Tr-periodic modules. In: Dlab and Gabriel [31], pp. 280–294
48.
go back to reference Nazarova, L.A.: Representations of quivers of infinite type (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 37, 752–791 (1973). English transl.,Math. USSR-Izv. 7(4) 749–792 (1973)CrossRef Nazarova, L.A.: Representations of quivers of infinite type (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 37, 752–791 (1973). English transl.,Math. USSR-Izv. 7(4) 749–792 (1973)CrossRef
52.
go back to reference Platzeck, M.I., Auslander, M.: Representation theory of hereditary Artin algebras. In: Gordon [38], pp. 389–424 Platzeck, M.I., Auslander, M.: Representation theory of hereditary Artin algebras. In: Gordon [38], pp. 389–424
58.
go back to reference Ringel, C.M.: The spectrum of a finite-dimensional algebra. In: F. van Oystaeyen (ed.) Ring Theory (Antewrp, 1978), Lecture Notes in Pure and Appl. Math., vol. 51, pp. 535–597. Dekker, New York (1979) Ringel, C.M.: The spectrum of a finite-dimensional algebra. In: F. van Oystaeyen (ed.) Ring Theory (Antewrp, 1978), Lecture Notes in Pure and Appl. Math., vol. 51, pp. 535–597. Dekker, New York (1979)
60.
go back to reference Roiter, A.V.: Unboundedness of the dimensions of the indecomposable representations of an algebra which has infinitely many indecomposable representations (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 32, 1275–1282 (1968)MathSciNet Roiter, A.V.: Unboundedness of the dimensions of the indecomposable representations of an algebra which has infinitely many indecomposable representations (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 32, 1275–1282 (1968)MathSciNet
62.
go back to reference Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 2. Tubes and Concealed Algebras of Euclidean Type, London Math. Soc. Stud. Texts, vol. 71. Cambridge Univ. Press, Cambridge (2007). DOI https://doi.org/10.1017/CBO9780511619212 Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 2. Tubes and Concealed Algebras of Euclidean Type, London Math. Soc. Stud. Texts, vol. 71. Cambridge Univ. Press, Cambridge (2007). DOI https://​doi.​org/​10.​1017/​CBO9780511619212​
63.
Metadata
Title
Auslander–Reiten Theory for Finite-Dimensional Algebras
Author
Piotr Malicki
Copyright Year
2018
DOI
https://doi.org/10.1007/978-3-319-74585-5_2

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