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Minimal singularities for representations of Dynkin quivers

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Commentarii Mathematici Helvetici

Abstract

We develop some reduction techniques for the study of singularities in orbit closures of finite dimensional modules. This enables us to classify all singularities occurring in minimal degenerations of representations of Dynkin quivers. They are all smoothly equivalent to the singularity at the zero-matrix inside thep×q-matrices of rank at most one.

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References

  • [1]Abeasis S andDel Fra A. Degenerations for the representations of an equioriented quiver of typeA m, J. of Algebra93 (1985), 376–412.

    Article  Google Scholar 

  • [2]Abeasis, S., del Fra, A. andKraft, H. The geometry of representations ofA m, Math. Ann.256 (1981), 401–418.

    Article  MathSciNet  Google Scholar 

  • Altman, A. andKleiman, S. Introduction to Grothendieck Duality Theory, Lecture Notes in Math.146, Springer Verlag (1970).

  • Auslander, M. andReiten, I. Modules determined by their composition factors, Illinois J. Math.29 (1985), 289–301.

    MathSciNet  Google Scholar 

  • Bongartz, K. Critical simply connected algebras, Man. Math.46 (1981), 117–136.

    Article  MathSciNet  Google Scholar 

  • Bongartz, K. A geometric version of the Morita equivalence, J. Algebra139 (1991), 159–179

    Article  MathSciNet  Google Scholar 

  • Bongartz, K. On degenerations and extensions of finite dimensional modules, to appear in Adv. Math., preprint 1-53 (1990).

  • Donkin, S. The normality of closures of conjugacy classes of matrices, Invent. math.101 (1990), 717–736.

    Article  MathSciNet  Google Scholar 

  • Gabriel, P. Unzerlegbare Darstellungen I, Man. Math.6, 71–103 (1972).

    Article  MathSciNet  Google Scholar 

  • Gabriel, P., Keller, B. andRoiter, A. V. Representations of finite-dimensional algebras, Encycl. Math. Sc.73 (1992), 1–176.

    MathSciNet  Google Scholar 

  • Kempken, G. Eine Darstellung des Köchers à k , Bonner Math. Schriften137 (1982), 1–159.

    MathSciNet  Google Scholar 

  • Kraft, H. Geometrische Methoden in der Invariantentheorie, Vieweg Verlag (1984).

  • Kraft, H. andProcesi, C. Closures of conjugacy classes of matrices are normal, Invent. math.53 (1979), 227–247.

    Article  MathSciNet  Google Scholar 

  • Kraft, H. andProcesi, C. Minimal singularities inGl n, Invent. math.62, 503–515 (1981).

    Article  MathSciNet  Google Scholar 

  • Kraft, H. andRiedtmann, C. Geometry of representations of quivers, LMS lecture notes116 (1985), 109–147.

    MathSciNet  Google Scholar 

  • Markolf, U. Entartungen von Darstellungen zu Dynkin-Köchern, Diplomarbeit Uni Wuppertal, (1990), 1–68.

  • Riedtmann, C. Degenerations for representations of quivers with relations, Ann. Sci. Ecole Norm. Sup.4 (1986), 275–301.

    MathSciNet  Google Scholar 

  • Ringel, C. M. Tame algebras and quadratic forms, Lecture Notes in Math. 1099, Springer Verlag (1984).

  • Slodowy, P. Simple singularities and simple algebraic groups, Lecture Notes in Math. 815, Springer Verlag (1980).

  • Serre, J. P. Espaces fibrés algébriques, Séminaire Chevalley 1958, 1–36.

  • Vinberg, E. B. andPopov, V. L. On a class of quasihomogeneous affine varieties, Math. USSR Izvestija6 (1972), 743–758.

    Article  Google Scholar 

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Bongartz, K. Minimal singularities for representations of Dynkin quivers. Commentarii Mathematici Helvetici 69, 575–611 (1994). https://doi.org/10.1007/BF02564505

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