Generalization of the Gell–Mann Decontraction Formula for sl(n, \( \mathbb{R} \)) and Its Applications in Affine Gravity

  • Igor Salom
  • Djordje Šijački
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 36)


The Gell–Mann Lie algebra decontraction formula was proposed as an inverse to the Inonu–Wigner contraction formula. We considered recently this formula in the content of the special linear algebras sl(n), of an arbitrary dimension. In the case of these algebras, the Gell–Mann formula is not valid generally, and holds only for some particular algebra representations. We constructed a generalization of the formula that is valid for an arbitrary irreducible representation of the sl(n) algebra. The generalization allows us to explicitly write down, in a closed form, all matrix elements of the algebra operators for an arbitrary irreducible representation, irrespectively whether it is tensorial or spinorial, finite or infinite dimensional, with or without multiplicity, unitary or nonunitary. The matrix elements are given in the basis of the Spin(n) subgroup of the corresponding SL(n, R) covering group, thus covering the most often cases of physical interest. The generalized Gell–Mann formula is presented, and as an illustration some details of its applications in the Gauge Affine theory of gravity with spinorial and tensorial matter manifields are given.


Irreducible Representation Affine Connection Reduce Matrix Element Shear Generator Dilaton Field 
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  1. 1.
    Berendt, G.: Acta Phys. Austriaca 25, 207 (1967)MATHGoogle Scholar
  2. 2.
    Harish-Chandra, Proc. Natl. Acad. Sci. 37, 170, 362, 366, 691 (1951)Google Scholar
  3. 3.
    Hazewinkel, M. (ed.): Encyclopaedia of Mathematics, Supplement I, p. 269. Springer, Berlin (1997)Google Scholar
  4. 4.
    Hehl, F.W., McCrea, J.D., Mielke, E.W., Ne’eman, Y.: Phys. Rep. 258, 1 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hermann, R.: Lie Groups for Physicists. W. A. Benjamin Inc, New York (1965)Google Scholar
  6. 6.
    Hermann, R.: Comm. Math. Phys. 2, 78 (1966)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Inönü, E., Wigner, E.P.: Proc. Natl. Acad. Sci. 39, 510 (1953)MATHCrossRefGoogle Scholar
  8. 8.
    Mackey, G.: Induced Representations of Groups and Quantum Mechanics. Benjamin, New York (1968)MATHGoogle Scholar
  9. 9.
    Mukunda, N.: J. Math. Phys. 10, 897 (1969)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ne’eman, Y., Šijački, Dj.: Ann. Phys. (N.Y.) 120, 292 (1979)Google Scholar
  11. 11.
    Ne’eman, Y., Šijački, Dj.: Int. J. Mod. Phys.  A2, 1655 (1987)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Salom, I.: Decontraction formula for \(sl(n, \mathbb{R})\) algebras and applications in theory of gravity. Ph.D. Thesis, Physics Department, University of Belgrade, 2011 (in Serbian)Google Scholar
  13. 13.
    Salom, I., Šijački, Dj.: Int. J. Geom. Met. Mod. Phys. 7, 455 (2010)Google Scholar
  14. 14.
    Salom, I., Šijački, Dj.: Lie theory and its applications in physics. AIPCP 1243, 191 (2010)Google Scholar
  15. 15.
    Salom, I., Šijački, Dj.: Int. J. Geom. Met. Mod. Phys. 8, 395 (2011)Google Scholar
  16. 16.
    Salom, I., Šijački, Dj.: [arXiv:math-ph/0904.4200v1]Google Scholar
  17. 17.
    Sankaranarayanan, A.: Nuovo Cimento 38, 1441 (1965)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Šijački, Dj.: Class. Quant. Gravit. 21, 4575 (2004)Google Scholar
  19. 19.
    Štoviček, P.: J. Math. Phys. 29, 1300 (1988)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Institute of PhysicsBelgradeSerbia

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