Letters in Mathematical Physics

, Volume 109, Issue 1, pp 113–133 | Cite as

Differential calculus on Jordan algebras and Jordan modules

  • Alessandro Carotenuto
  • Ludwik Dąbrowski
  • Michel Dubois-VioletteEmail author


Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra and provide a complete characterization of the theory of connections for free Jordan modules.


Jordan algebras Jordan modules Differential calculus Connections 

Mathematics Subject Classification

17Cxx 97140 


  1. 1.
    Albert, A.A.: On a certain algebra of quantum mechanic (second series). Ann. Math. 35, 65–72 (1934)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baez, J.: The octonions. arXiv:math/0105155 (2002)
  3. 3.
    Chamseddine, A., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Connes, A.: Noncommutative differential geometry. Publ. IHES 62, 257–360 (1986)Google Scholar
  5. 5.
    Connes, A., Lott, J.: Particles models and noncommutative geometry. Nucl. Phys. B18(Suppl), 29–47 (1990)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Connes, A.: Noncommutative Geometry. Accademic Press, New York (1994)zbMATHGoogle Scholar
  7. 7.
    Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  8. 8.
    Dubois-Violette, M., Kerner, R., Madore, J.: Gauge bosons in a non-commutative geometry. Phys. Lett. B217, 485–488 (1989)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Dubois-Violette, M.: Noncommutative differential geometry, quantum mechanics and gauge theory. In: Bartocci, C., Bruzzo, U., Cianci, R. (eds.) Differential Geometric Methods in Theoretical Physics. Lecture Notes in Physics, vol. 375. Springer, Berlin (1991)Google Scholar
  10. 10.
    Dubois-Violette, M.: Lectures on graded differential geometry and noncommutative geometry. In: Maeda, Y., et al. (eds.) Noncommutative Differential Geometry and Its Applications to Physics, pp. 245–306. Kluwer Accademic Pubblishers, Shonan (2001)CrossRefGoogle Scholar
  11. 11.
    Dubois-Violette, M.: Exceptional quantum geometry and particle physics. Nucl. Phys. B912, 426–449 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eilenberg, S.: Extensions of general algebras. Ann. Soc. Math. Pol. 21, 125–134 (1948)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gunaydin, M., Piron, C., Ruegg, H.: Moufang plane and octionic quantum mechanics. Commun. Math. Phys. 61, 69–85 (1978)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Harris, B.: Derivations of Jordan algebras. Pac. J. Math. 9, 495–512 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Huebschmann, J.: Lie–Rineheart algebras. Gerstenhaber algebras and Batalin–Vilkovisky algebras. Ann. Inst. Fourier Grenoble 48, 425–440 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iordanescu, R.: Jordan Structures in Analysis, Geometry and Physics. Editura Academiei Române, Bucharest (2009)zbMATHGoogle Scholar
  17. 17.
    Jacobson, N.: General representation theory of Jordan algebras. Trans. Am. Math. Soc. 70, 509–530 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jacobson, N.: Structure and Representations of Jordan Algebras. American Mathematical Society, Providence (1968)CrossRefzbMATHGoogle Scholar
  19. 19.
    Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kac, V.G.: Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras. Commun. Algebra 5, 1375–1400 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kashuba, I., Osvienko, S., Shestakov, I.: Representation type of Jordan algebras. Adv. Math. 226, 1–35 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Koszul, J.L.: Fibre Bundles and Differential Geometry. Tata Institute of Fundamental Research, Bombay (1960)Google Scholar
  23. 23.
    McCrimmon, K.: A Taste of Jordan Algebras. Springer, Berlin (2004)zbMATHGoogle Scholar
  24. 24.
    Rinehart, G.: Differential forms for general commutative algebras. Trans. Am. Math. Soc. 108, 195–222 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    van Suijlekom, W.: Noncommutative Geometry and Particle Physics. Springer, Berlin (2014)zbMATHGoogle Scholar
  26. 26.
    Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys. 122, 125–170 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wulfsohn, A.: Tensor product of Jordan algebras. Can. J. Math. XXVIII, 60–74 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Yokota, I.: Exceptional Lie groups. arXiv:0902.0431 (2009)
  29. 29.
    Zhelyabin, V.N., Shestakov, I.P.: Simple special Jordan superalgebras with associative even part. Sib. Math. J. 45, 860–882 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Alessandro Carotenuto
    • 1
  • Ludwik Dąbrowski
    • 1
  • Michel Dubois-Violette
    • 2
    Email author
  1. 1.Scuola Internazionale Superiore di Studi Avanzati (SISSA)TriesteItaly
  2. 2.Laboratoire de Physique Théorique, CNRSUniversité Paris-Sud, Université Paris-SaclayOrsayFrance

Personalised recommendations