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Solvable Algebras and the Tits Satake Projection

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

Abstract

In this chapter we are going to develop the details of a theory pertaining to Lie Algebras which, although it has its roots in mathematical work of the 1960s (Satake in Ann Math 71:77–110, 1960, [1], Tits in Algebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics, Boulder, Colorado, 1965, p. 33–62 [2], Borel, Tits in Groupes réductifs. Publications Mathémathiques de l’IHES 27: 55–151, 1965; Compléments à l’article 41:253–276 1972, [3]), contributed by two great algebrists, Jacques Tits and Ichiro Satake (see Fig. 5.1), yet fully revealed its profound significance for Geometry and Physics only much later, by the end of the XXth century, and within the context of supergravity.

References

  1. 1.
    I. Satake, On representations and compactifications of symmetric riemannian spaces. Ann. Math. 71, 77–110 (1960). Second SeriesMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Tits, Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups (Proceedings of Symposia in Pure Mathematics, Boulder, Colorado, 1965), p. 33–62Google Scholar
  3. 3.
    A. Borel, J. Tits, Groupes réductifs. Publications Mathémathiques de l’IHES 27, 55–151 (1965); Compléments à l’article 41, 253–276 (1972)Google Scholar
  4. 4.
    B. de Wit, A. Van Proeyen, Broken sigma model isometries in very special geometry. Phys. Lett. B 293, 94–99 (1992)ADSCrossRefGoogle Scholar
  5. 5.
    B. de Wit, F. Vanderseypen, A. Van Proeyen, Symmetry structure of special geometries. Nucl. Phys. B 400, 463–524 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    B. de Wit, A. Van Proeyen, Isometries of special manifolds (1995)Google Scholar
  7. 7.
    D.V. Alekseevsky, V. Cortes, C. Devchand, A. Van Proeyen, Polyvector super Poincare algebras. Commun. Math. Phys. 253, 385–422 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    L. Andrianopoli, R. D’Auria, S. Ferrara, P. Fré, M. Trigiante, R-R scalars, U-duality and solvable Lie algebras. Nucl. Phys. B 496, 617–629 (1997), arXiv:hep-th/9611014
  9. 9.
    L. Andrianopoli, R. D’Auria, S. Ferrara, P. Fré, R. Minasian, M. Trigiante, Solvable Lie algebras in type IIA, type IIB and M-theories. Nucl. Phys. B 493, 249–277 (1997), arXiv:hep-th/9612202
  10. 10.
    M. Trigiante, Dualities in supergravity and solvable Lie algebras, Ph.D. thesis, Swansea University, 1998Google Scholar
  11. 11.
    V. Belinsky, I. Khalatnikov, E. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology. Adv. Phys. 19, 525–573 (1970)ADSCrossRefGoogle Scholar
  12. 12.
    V. Belinsky, I. Khalatnikov, E. Lifshitz, A general solution of the einstein equations with a time singularity. Adv. Phys. 31, 639–667 (1982)ADSCrossRefGoogle Scholar
  13. 13.
    J.K. Erickson, D.H. Wesley, P.J. Steinhardt, N. Turok, Kasner and mixmaster behavior in universes with equation of state \(w\ge 1\). Phys. Rev. D 69, 063514 (2004)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Demaret, M. Henneaux, P. Spindel, Nonoscillatory behavior in vacuum Kaluza–Klein cosmologies. Phys. Lett. 164B, 27–30 (1985)ADSCrossRefGoogle Scholar
  15. 15.
    J. Demaret, J.L. Hanquin, M. Henneaux, P. Spindel, A. Taormina, The fate of the mixmaster behavior in vacuum inhomogeneous Kaluza–Klein cosmological models. Phys. Lett. B 175, 129–132 (1986)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    J. Demaret, Y. De Rop, M. Henneaux, Chaos in nondiagonal spatially homogeneous cosmological models in space-time dimensions \(\le \) 10. Phys. Lett. B 211, 37–41 (1988)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    T. Damour, M. Henneaux, B. Julia, H. Nicolai, Hyperbolic Kac–Moody algebras and chaos in Kaluza–Klein models. Phys. Lett. B 509, 323–330 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    T. Damour, S. de Buyl, M. Henneaux, C. Schomblond, Einstein billiards and overextensions of finite dimensional simple Lie algebras. JHEP 08, 030 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    T. Damour, M. Henneaux, H. Nicolai, Cosmological billiards. Class. Quantum Gravity 20, R145–R200 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    S. de Buyl, M. Henneaux, B. Julia, L. Paulot, Cosmological billiards and oxidation. Fortsch. Phys. 52, 548–554 (2004). [PoSjhw2003, 015 (2003)]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J. Brown, O.J. Ganor, C. Helfgott, M theory and E(10): billiards, branes, and imaginary roots. JHEP 08, 063 (2004)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    F. Englert, M. Henneaux, L. Houart, From very-extended to overextended gravity and M-theories. JHEP 02, 070 (2005)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    T. Damour, Cosmological singularities, Einstein billiards and Lorentzian Kac–Moody algebras, in Miami Waves 2004: Conference on Geometric Analysis, Nonlinear Wave Equations and General Relativity Coral Gables, Florida, 4-10 January 2004, (2005). [J. Hyperbol. Diff. Equat. (2005)]Google Scholar
  24. 24.
    T. Damour, Poincare, relativity, billiards and symmetry, in Proceedings, Symposium Henri Poincare, Brussels, Belgium, 8-9 October 2004, (2005), p. 149Google Scholar
  25. 25.
    M. Henneaux, B. Julia, Hyperbolic billiards of pure D \(=\) 4 supergravities. JHEP 05, 047 (2003)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    A. Keurentjes, The group theory of oxidation. Nucl. Phys. B 658, 303–347 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    A. Keurentjes, The group theory of oxidation 2: cosets of nonsplit groups. Nucl. Phys. B 658, 348–372 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    P. Fré, V. Gili, F. Gargiulo, A.S. Sorin, K. Rulik, M. Trigiante, Cosmological backgrounds of superstring theory and solvable algebras: oxidation and branes. Nucl. Phys. B 685, 3–64 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    P. Fré, A.S. Sorin, Integrability of supergravity billiards and the generalized Toda lattice equation. Nucl. Phys. B 733, 334–355 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    P. Fre, A.S. Sorin, The Weyl group and asymptotics: all supergravity billiards have a closed form general integral. Nucl. Phys. B 815, 430–494 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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).  https://doi.org/10.1016/j.nuclphysb.2005.10.023, arXiv:hep-th/0507256
  32. 32.
    P. Fre, F. Gargiulo, J. Rosseel, K. Rulik, M. Trigiante, A. Van Proeyen, Tits–Satake projections of homogeneous special geometries. Class. Quantum Gravity 24, 27–78 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    P. Fré, A. Sorin, Supergravity black holes and billiards and liouville integrable structure of dual borel algebras. JHEP 03, 066 (2010)ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    P. Fré, A. Sorin, The integration algorithm for nilpotent orbits of \(g/h^\star \) lax systems: for extremal black holes (2009)Google Scholar
  35. 35.
    W. Chemissany, J. Rosseel, M. Trigiante, T. Van Riet, The full integration of black hole solutions to symmetric supergravity theories. Nucl. Phys. B 830, 391 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    M. Gaillard, Z. Bruno, Duality rotations for interacting fields. Nucl. Phys. B 193, 221 (1981)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    P. Breitenlohner, D. Maison, G.W. Gibbons, Four-dimensional black holes from Kaluza–Klein theories. Commun. Math. Phys. 120, 295 (1988)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    M. Gunaydin, A. Neitzke, B. Pioline, A. Waldron, Bps black holes, quantum attractor flows and automorphic forms. Phys. Rev. D 73, 084019 (2006)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    B. Pioline, Lectures on black holes, topological strings and quantum attractors. Class. Quantum Gravity 23, S981 (2006)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    M. Gunaydin, A. Neitzke, B. Pioline, A. Waldron, Quantum attractor flows. JHEP 0709, 056 (2007)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    D. Gaiotto, W. Li, M. Padi, Non-supersymmetric attractor flow in symmetric spaces. JHEP 12, 093 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    E. Bergshoeff, W. Chemissany, A. Ploegh, M. Trigiante, T. Van Riet, Generating geodesic flows and supergravity solutions. Nucl. Phys. B 812, 343 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    G. Bossard, H. Nicolai, K. Stelle, Universal bps structure of stationary supergravity solutions. JHEP 0907, 003 (2009)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    P. Frè, A. Sorin, M. Trigiante, Integrability of supergravity black holes and new tensor classifiers of regular and nilpotent orbits. JHEP 1204, 015 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    P. Frè, A. Sorin, M. Trigiante, Black hole nilpotent orbits and tits satake universality classes (2011)Google Scholar
  46. 46.
    P. Fre’, F. Gargiulo, K. Rulik, M. Trigiante, The general pattern of Kac Moody extensions in supergravity and the issue of cosmic billiards. Nucl. Phys. B 741, 42–82 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Pietra MarazziItaly

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