Abstract
The main purpose of this lecture note is to provide a quick introduction to quivers and their representations. In particular, as there already exists several introductory and complete texts on quivers, the author tries motivating the reader to develop the theory by showing several concrete examples.
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References
I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras, vol. 1. Techniques of Representation Theory, London Mathematical Society Student Texts, vol. 65 (Cambridge University Press, Cambridge, 2006).
D.J. Benson, Representations and Cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, 2nd edn. (Cambridge University Press, Cambridge, 1998). Basic Representation Theory of Finite Groups and Associative Algebras
I.N. Bernšteĭn, I.M. Gel’fand, V.A. Ponomarev, Coxeter functors, and Gabriel’s theorem. Uspehi Mat. Nauk 28(2(170)), 19–33 (1973)
M. Brion, Representations of quivers, in Geometric Methods in Representation Theory. I, Sémin. Congr., vol. 24 (Soc. Math. France, Paris, 2012), pp. 103–144
J. Dieudonné, Sur la réduction canonique des couples de matrices. Bull. Soc. Math. Fr. 74, 130–146 (1946)
V. Dlab, C.M. Ringel, Indecomposable representations of graphs and algebras. Mem. Amer. Math. Soc. 6(173), v+57 (1976)
P. Donovan, M.R. Freislich, The Representation Theory of Finite Graphs and Associated Algebras, Carleton Mathematical Lecture Notes, No. 5 (Carleton University, Ottawa, Ont., 1973)
P. Gabriel, Unzerlegbare Darstellungen. I. Manuscr. Math. 6, 71–103 (1972); correction, Manuscr. Math. 6, 309 (1972)
P. Gabriel, Représentations indécomposables, in Séminaire Bourbaki, 26e année (1973/1974), Exp. No. 444, Lecture Notes in Math., vol. 431 (Springer, Berlin, 1975), pp. 143–169
D. Happel, D. Zacharia, Algebras of finite global dimension, in Algebras, Quivers and Representations, Abel Symp., vol. 8 (Springer, Heidelberg, 2013), pp. 95–113
K. Hiroe, On additive Deligne-Simpson problems, in Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers, Algorithms Comput. Math. (Springer, Berlin, 2019)
V.G. Kac, Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56(1), 57–92 (1980)
V.G. Kac, Infinite root systems, representations of graphs and invariant theory. II. J. Algebra 78(1), 141–162 (1982)
V.G. Kac, Root systems, representations of quivers and invariant theory, in Invariant Theory (Montecatini, 1982), Lecture Notes in Math., vol. 996 (Springer, Berlin, 1983), pp. 74–108
V.G. Kac, Infinite-dimensional Lie Algebras, 3rd edn. (Cambridge University Press, Cambridge, 1990)
Y. Kimura, Introduction to quiver varieties, in Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers, Algorithms Comput. Math. (Springer, Berlin, 2019)
H. Kraft, C. Riedtmann, Geometry of representations of quivers, in Representations of Algebras (Durham, 1985), London Math. Soc. Lecture Note Ser., vol. 116 (Cambridge Univ. Press, Cambridge, 1986), pp. 109–145
L. Kronecker, Algebraische Reduction der Scharen bilinearer Formen. Berl. Ber. 1890, 1225–1237 (1890)
L.A. Nazarova, Representations of quivers of infinite type. Izv. Akad. Nauk SSSR Ser. Mat. 37, 752–791 (1973)
C.M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics, vol. 1099 (Springer-Verlag, Berlin, 1984)
R. Schiffler, Quiver representations. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (Springer, Cham, 2014)
D. Yamakawa, Applications of quiver varieties to moduli spaces of connections on \(\mathbb{P}^1\), in Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers, Algorithms Comput. Math. (Springer, Berlin, 2019)
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Iohara, K. (2020). Introduction to Representations of Quivers. In: Iohara, K., Malbos, P., Saito, MH., Takayama, N. (eds) Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers. Algorithms and Computation in Mathematics, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-030-26454-3_6
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