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Generalized Dynkin Diagrams and Root Systems and Their Folding

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Topological Field Theory, Primitive Forms and Related Topics

Part of the book series: Progress in Mathematics ((PM,volume 160))

Abstract

Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural “Coxeter element.” The folding of these graphs and groups is also discussed, using the theory of C-algebras.

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References

  1. V.I. Arnold, and S.M. Gusein-Zade, A.N. Varchenko, Singularities of Differentiable Maps, Birkäuser, Basel 1985.

    Book  MATH  Google Scholar 

  2. N. Bourbaki, Groupes et Algèbres de Lie, chap. 4–6, Masson 1981; J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univ. Pr. 1990.

    Google Scholar 

  3. E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings (1984).

    Google Scholar 

  4. S. Berman, Y.S. Lee and R.V. Moody, The Spectrum of a Coxeter Transformation, Affine Coxeter Transformations, and the Defect Map, J. Algebra 121 (1989) 339–357.

    Article  MathSciNet  MATH  Google Scholar 

  5. H.S.M. Coxeter, Discrete Groups Generated by Reflections, Ann. Math. 35 (1934) 588–621.

    Article  MathSciNet  Google Scholar 

  6. S. Cecotti and C. Vafa, Nucl. Phys. B367 (1991) 359–461; Comm. Math. Phys. 158 (1993) 569-644.

    Article  MathSciNet  Google Scholar 

  7. B. Dubrovin, Differential Geometry of the space of orbits of a reflection group, hep-th/9303152; Springer Lect. Notes in Math. 1620 (1996) 120–348, hep-th/9407018; B. Dubrovin and Y. Zhang, Extended affine Weyl groups and Frobenius manifolds, hep-th/9611200.

    MathSciNet  Google Scholar 

  8. P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer Verlag 1997, and further references therein.

    Google Scholar 

  9. P. Di Francesco and J.-B. Zuber, SU(N) Lattice Integrable Models Associated with Graphs, Nucl. Phys. B338 (1990) 602–646.

    Article  Google Scholar 

  10. P. Di Francesco and J.-B. Zuber, SU(N) Lattice Integrable Models and Modular Invariance, in Recent Developments in Conformal Field Theories, Trieste Conference, 1989, S. Randjbar-Daemi, E. Sezgin and J.-B. Zuber eds., World Scientific 1990; P. Di Francesco, Int. J. Mod. Phys. A7 (1992) 407-500.

    Google Scholar 

  11. W.M. Fairbairn, T. Fulton and W.H. Klink, Finite and Disconnected Subgroups of SU3and their Application to the Elementary-Particle Spectrum, J. Math. Phys 5 (1964) 1038–1051.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Fendley and P. Ginsparg, Non-Critical Orbifolds, Nucl. Phys. B324 (1989) 549–580.

    Article  MathSciNet  Google Scholar 

  13. W. Fulton and J. Harris, Representation Theory, Springer Verlag.

    Google Scholar 

  14. D. Gepner, Fusion Rings and Geometry, Comm. Math. Phys. 141 (1991) 381–4

    Article  MathSciNet  MATH  Google Scholar 

  15. F. Goodman, P. de la Harpe and V.F.R. Jones, Coxeter graphs and towers of algebras, 14 MSRI Publ., Springer (1989).

    Book  MATH  Google Scholar 

  16. S.M. Gusein-Zade and A. Varchenko, Verlinde algebras and the intersection form on vanishing cycles, hep-th/9610058.

    Google Scholar 

  17. V.G. Kac and D.H. Peterson, Infinite Dimensional Lie Algebras, Theta Functions and Modular Forms, Adv. Math. 53 (1984) 125–263.

    Article  MathSciNet  MATH  Google Scholar 

  18. B. Kostant, The McKay Correspondence, the Coxeter Element and Representation Theory, Société Mathématique de France, Astérisque (1988) 209–255.

    Google Scholar 

  19. I.K. Kostov, Free Field Representation of the AnCoset Models on the Torus, Nucl. Phys. B 300 [FS22] (1988) 559–587.

    Article  MathSciNet  Google Scholar 

  20. J. MacKay, Graphs, Singularities, and Finite Groups, Proc. Symp. Pure Math. 37 (1980) 183–186.

    Google Scholar 

  21. R.V. Moody and J. Patera, Quasicrystals and Icosians, J. Phys. A 26 (1993) 2829–2853.

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Moore and N. Seiberg, Classical and Quantum Field Theory, Comm. Math. Phys. 123 177–254 (1989); Lectures on RCFT, in Superstrings 89, proceedings of the 1989 Trieste spring school, M. Green et al. eds, World Scientific 1990.

    Google Scholar 

  23. S.G. Naculich and H.J. Schnitzer, Superconformai coset equivalence from level-rank duality, hep-th/9705149.

    Google Scholar 

  24. A. Ocneanu, communication at the Workshop Low Dimensional Topology, Statistical Mechanics and Quantum Field Theory, Fields Institute, Waterloo, Ontario, April 26–30, 1995.

    Google Scholar 

  25. V. Pasquier, Operator Content of the ADE lattice models, J. Phys. A20 5707–5717 (1987); Thèse d’Etat, Orsay, 1988.

    MathSciNet  Google Scholar 

  26. V.B. Petkova, private communication and to appear.

    Google Scholar 

  27. V.B. Petkova and J.-B. Zuber, From CFT to Graphs, Nucl. Phys. B B463 (1996) 161–193: hep-th/9510175; Conformal Field Theory and Graphs, hep-th/9701103.

    Article  MathSciNet  Google Scholar 

  28. K. Saito, Extended Root Systems I (Coxeter transformation), Publ. RIMS, Kyoto Univ. 21 (1985) 75–179, 26 (1990) 15-78; K. Saito and T. Takebayashi, preprint RIMS-1089.

    Article  MATH  Google Scholar 

  29. O.P. Shcherbak, Wavefronts and Reflection Groups, Russ. Math. Surveys 43:3 (1988) 149–194.

    Article  MathSciNet  Google Scholar 

  30. J. Steenbrink, Intersection From for Quasi-Homogeneous Singularities, Compositio Math. 34 (1977) 211–223.

    MathSciNet  MATH  Google Scholar 

  31. A.N. Varchenko, Zeta-Function of Monodromy and Newton’s Diagram, Inv. Math. 37 (1976) 253–262.

    Article  MathSciNet  MATH  Google Scholar 

  32. E. Verlinde, Fusion Rules and Modular Transformations in 2-D Conformal Field Theory, Nucl. Phys. B300 [FS22] (1988) 360–376.

    Article  MathSciNet  Google Scholar 

  33. J. Sekiguchi and T. Yano, A Note on the Coxeter Group of Type H3, Sci. Rep. Saitama Univ. IX (1980) 33–44; T. Yano, ibid. 61-70.

    Google Scholar 

  34. J.-B. Zuber, Graphs and Reflection Groups, Comm. Math. Phys. 179 (1996) 265–294.

    Article  MathSciNet  MATH  Google Scholar 

  35. J.-B. Zuber, Conformal, Integrable and Topological Theories, Graphs and Coxeter Groups, in XIth International Congress of Mathematical Physics, Paris July 1994, D. Iagolnitzer ed., International Press 1995, p. 674–689, hep-th/9412202.

    Google Scholar 

  36. J.-B. Zuber, C-algebras and their applications to reflection groups and conformai field theories, proceedings of the RIMS Symposium, Kyoto, 16–19 December 1996, hep-th/9707034.

    Google Scholar 

  37. J.-B. Zuber, On Dubrovin Topological Field Theories, Mod. Phys. Lett A8 (1994) 749–760.

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Service de Physique Théorique, CEA Saclay, F-91191, Gif sur Yvette Cedex, France

    Jean-Bernard Zuber

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  1. Jean-Bernard Zuber
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Editors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-01, Japan

    Masaki Kashiwara  & Kyoji Saito  & 

  2. Department of Mathematical Sciences, University of Tokyo, Tokyo, 153, Japan

    Atsushi Matsuo

  3. Department of Mathematics, Osaka University, Toyonaka-city, 560, Japan

    Ikuo Satake

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Zuber, JB. (1998). Generalized Dynkin Diagrams and Root Systems and Their Folding. In: Kashiwara, M., Matsuo, A., Saito, K., Satake, I. (eds) Topological Field Theory, Primitive Forms and Related Topics. Progress in Mathematics, vol 160. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0705-4_16

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  • DOI: https://doi.org/10.1007/978-1-4612-0705-4_16

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6874-1

  • Online ISBN: 978-1-4612-0705-4

  • eBook Packages: Springer Book Archive

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