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Published in:
Representations of Quivers Read chapter Read first chapter

2014 | OriginalPaper | Chapter

8. Quadratic Forms and Gabriel’s Theorem

Author : Ralf Schiffler

Published in: Quiver Representations

Publisher: Springer International Publishing

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Abstract

The main goal of this chapter is to prove that the number of isoclasses of indecomposable representations of a connected quiver Q is finite if and only if Q is of Dynkin type \(\mathbb{A}, \mathbb{D}\) or \(\mathbb{E}\). The proof we are presenting uses the classification of positive definite integral quadratic forms associated to graphs and also a little algebraic geometry. For a different proof, using tilting theory, see [8, VII.5].

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previous chapter Auslander–Reiten Theory
Literature
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Maurice Auslander, Idun Reiten, and SmaløSverre O., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422 (96c:16015)
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I. N. Bernšteĭn, I. M. Gel’fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33. MR 0393065 (52 #13876)
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V. G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), no. 1, 57–92. MR 557581 (82j:16050)
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Metadata
Title
Quadratic Forms and Gabriel’s Theorem
Author
Ralf Schiffler
Copyright Year
2014
Book
Quiver Representations

Print ISBN: 978-3-319-09203-4

Electronic ISBN: 978-3-319-09204-1

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-09204-1_8

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