Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On minimal disjoint degenerations for preprojective representations of quivers

Authors: Klaus Bongartz and Thomas Fritzsche
Journal: Math. Comp. 72 (2003), 2013-2042
MSC (2000): Primary 16G20, 14L30
Published electronically: February 3, 2003
MathSciNet review: 1986819
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We derive a root test for degenerations as described in the title. In the case of Dynkin quivers this leads to a conceptual proof of the fact that the codimension of a minimal disjoint degeneration is always one. For Euclidean quivers, it enables us to show a periodic behaviour. This reduces the classification of all these degenerations to a finite problem that we have solved with the aid of a computer. It turns out that the codimensions are bounded by two. Somewhat surprisingly, the regular and preinjective modules play an essential role in our proofs.

References [Enhancements On Off] (What's this?)

  • 1. Abeasis, S., del Fra, A.: Degenerations for the representations of a quiver of type $A_{m}$, J.Algebra 93 (1985), 376-412. MR 86j:16028
  • 2. Auslander, M., Reiten, I., Smalø, S.O.: Representation theory of artin algebras, Cambridge Studies in Advanced Mathematics Vol. 3, Cambridge University Press. MR 96c:16015
  • 3. Bender, J., Bongartz, K.: Minimal singularities in orbits of matrix pencils, to appear in Linear Algebra Appl.
  • 4. Bongartz, K: On degenerations and extensions of finite dimensional modules, Math. 121 (1996), 245-287. MR 98e:16012
  • 5. Bongartz, K.: Minimal singularities for representations of Dynkin quivers, Comment. Math. Helv. 69(1994), 575-611. MR 96f:16016
  • 6. Bourbaki, N.: Groupes et algebres de Lie, chapitres 4,5 et 6, Hermann, Paris 1968. MR 39:1590
  • 7. Dlab, V., Ringel, C.M.: Indecomposable representations of graphs and algebras, Memoirs Amer. Math. Soc., Vol.173, 1976. MR 56:5657
  • 8. Fritzsche, T.: Minimale disjunkte Entartungen zwischen präprojektiven Darstellungen Euklidischer Kocher, Diplomarbeit BUGH Wuppertal, 2001.
  • 9. Kac, V.G.: Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), 57-92. MR 82j:16050
  • 10. Markolf, U.: Entartungen von Moduln über darstellungsgerichteten Algebren, Diplomarbeit 1990, Bergische Universität Wuppertal.
  • 11. Moody, R.V.: Euclidean Lie algebras, Can. J. Math. 21 (1969), 1432-1454. MR 41:287
  • 12. Riedtmann, C.: Degenerations for representations of quivers with relations, Ann. Sci. Ecole Norm. Sup.(4) 19 (1986), 275-301. MR 88b:16051
  • 13. Ringel, C.M.: Tame algebras and integral quadratic forms, Lecture Notes in Mathematics 1099, Springer 1984. MR 87f:16027

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 16G20, 14L30

Retrieve articles in all journals with MSC (2000): 16G20, 14L30

Additional Information

Klaus Bongartz
Affiliation: FB Mathematik BUGH Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany

Thomas Fritzsche
Affiliation: FB Mathematik BUGH Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany

Received by editor(s): March 1, 2001
Received by editor(s) in revised form: March 4, 2002
Published electronically: February 3, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society