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Finite Dimensional Lie Algebras in Singularities

In: Handbook of Geometry and Topology of Singularities I

Author

Listed:
  • José Luis Cisneros Molina

    (Universidad Nacional Autónoma de México, Unidad Cuernavaca, Instituto de Matemáticas
    Laboratorio Solomon Lefschetz, Unidad Mixta Internacional 2001 CNRS)

  • Meral Tosun

    (Galatasaray University, Çirağan Caddesi)

Abstract

Complex simple Lie algebras with simply laced root systems are classified by Dynkin diagrams of type An, Dn, E6, E7, and E8. Also the dual graphs of the minimal resolution of Kleinian singularities are precisely the same aforementioned Dynkin diagrams. In this work, we recall the basic definitions and some results of the theory of complex Lie algebras and of Kleinian singularities, in order to present a relation between finite dimensional complex simple Lie algebras and the Kleinian singularities, given by a theorem by Brieskorn. We also present the extension of Brieskorn’s theorem to the simple elliptic singularity D ~ 5 $$\tilde {D}_5$$ .

Suggested Citation

  • José Luis Cisneros Molina & Meral Tosun, 2020. "Finite Dimensional Lie Algebras in Singularities," Springer Books, in: José Luis Cisneros Molina & Dũng Tráng Lê & José Seade (ed.), Handbook of Geometry and Topology of Singularities I, chapter 0, pages 541-587, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-53061-7_10
    DOI: 10.1007/978-3-030-53061-7_10
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