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Introduction to Quiver Varieties

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Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers

Part of the book series: Algorithms and Computation in Mathematics ((AACIM,volume 28))

Abstract

Quiver representations and Kac–Moody Lie algebras. The interaction between quiver representations and Kac–Moody Lie algebras has an origin in Gabriel’s theorem. Gabriel [15] classified the quivers which are finite representation types and showed the existence of the bijection between the set of isomorphism classes of indecomposable representations of Dynkin quivers Q and the set of positive roots of the corresponding simply laced Lie algebra \({\mathfrak {g}}_{Q}\) via dimension vectors. Bernšteĭn–Gel’fand–Ponomarev [2] gave a proof of Gabriel’s theorem using reflection functors and Coxeter functors. Using the theory of species introduced by Gabriel [16], Dlab–Ringel [11] extended Gabriel’s theorem to finite dimensional hereditary algebras over arbitrary fields. The classification of the indecomposable representations of affine quivers was studied by Weierstrass, Kronecker, Gel’fand–Ponomarev, Donovan–Freislich, Nazarova and Dlab–Ringel [12].

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Notes

  1. 1.

    ALE stands for asymptotically locally Euclidean.

  2. 2.

    ADHM stands for Atiyah–Drinfeld–Hitchin–Manin.

  3. 3.

    S stands for Seshadri.

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Authors and Affiliations

  1. Faculty of Liberal Arts and Sciences, Osaka Prefecture University, Osaka, 599-8531, Japan

    Yoshiyuki Kimura

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  1. Yoshiyuki Kimura
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Editors and Affiliations

  1. Université Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, France

    Kenji Iohara

  2. Université Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, France

    Philippe Malbos

  3. Center for Mathematical and Data Sciences, Department of Mathematics, Graduate School of Science, Kobe University, Kobe, Japan

    Masa-Hiko Saito

  4. Department of Mathematics, Graduate School of Science, Kobe University, Kobe, Japan

    Nobuki Takayama

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Kimura, Y. (2020). Introduction to Quiver Varieties. 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_7

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