Finite Groups and Lie Algebras: The ADE Classification and Beyond

  • Pietro Giuseppe FréEmail author
Part of the Theoretical and Mathematical Physics book series (TMP)


The geometrical structures, mostly motivated by supergravity, that are considered in this book are strongly related with the theory of symmetric spaces and of Lie Algebras, the exceptional ones being of utmost relevance in this context.


  1. 1.
    P.G. Fré, A. Fedotov, Groups and Manifolds: Lectures for Physicists with Examples in Mathematica (De Gruyter, Berlin, 2018)zbMATHGoogle Scholar
  2. 2.
    G. James, M.W. Liebeck, Representations and Characters of Groups (Cambridge University Press, Cambridge, 2001)Google Scholar
  3. 3.
    A. Hurwitz, Uber algebraische gebilde mit eindeutigen transformationen in sich. Mathematische Annalen 41, 3 (1893)MathSciNetzbMATHGoogle Scholar
  4. 4.
    F. Klein, Ueber die transformation siebenter ordnung der elliptischen functionen. Mathematische Annalen 14, 3 (1878)MathSciNetGoogle Scholar
  5. 5.
    G. Fano, Sui postulati fondamentali della geometria proiettiva, Giornale di Matematiche, 30 (1892)Google Scholar
  6. 6.
    R.C. King, F. Toumazet, B.G. Wybourne, A finite subgroup of the exceptional Lie group G2. J. Phys. A. Math. Gen. 32, 8527–8537 (1999)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    P. Fré, Supersymmetric M2-branes with Englert fluxes, and the simple group PSL(2, 7). Fortsch. Phys. 64, 425–462 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    P. G. Fré, A Conceptual History of Symmetry from Plato to the Superworld (Springer, Berlin, 2018)Google Scholar
  9. 9.
    P. Fre’, M. Trigiante, Twisted tori and fluxes: a no go theorem for Lie groups of weak G(2) holonomy. Nucl. Phys. B 751, 343–375 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    C. Luhn, S. Nasri, P. Ramond, Simple finite non-Abelian flavor groups. J. Math. Phys. 48, 123519 (2007). arXiv:0709.1447 [hep-th]
  11. 11.
    J.E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer, Berlin, 1972)Google Scholar
  12. 12.
    P. Fré, F. Gargiulo, K. Rulik, Cosmic billiards with painted walls in non-maximal supergravities: a worked out example. Nucl. Phys. B 737, 1–48 (2006), [hep-th/0507256]
  13. 13.
    S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, Cambridge, 1962)Google Scholar
  14. 14.
    N. Jacobson, Lie Algebras (Courier Corporation, 1979)Google Scholar
  15. 15.
    V.S. Varadarajan, Lie Groups, Lie Algebras, and their Representations (Springer Science & Business Media, Berlin, 2013)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Pietra MarazziItaly

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