Skip to main content
SpringerLink
Log in

Indecomposable representations and oscillator realizations of the exceptional Lie algebra G2

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

In this paper various representations of the exceptional Lie algebra G2 are investigated in a purely algebraic manner, and multi-boson/multi-fermion realizations are obtained. Matrix elements of the master representation, which is defined on the space of the universal enveloping algebra of G2, are explicitly determined. From this master representation, different indecomposable representations defined on invariant subspaces or quotient spaces with respect to these invariant subspaces are discussed. Especially, the Verma module are investigated in detail to construct the Dyson-Maleev-like realization of G2. After obtaining explicit forms of all twelve extremal vectors of the Verma module with the dominant highest weight \(\Lambda \) , representations with their respective highest weights related to \(\Lambda\) are systematically discussed. Moreover, this method can be used to construct fermion realizations from the irreducible representations, without referring to the Lie algebra chains, and a three-fermion realization is constructed as an example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Racah, Phys. Rev. 76, 1352 (1949)

    Article  ADS  MATH  Google Scholar 

  2. H.A. Jahn, Proc. R. Soc. London A 201, 516 (1950)

    Article  ADS  MATH  Google Scholar 

  3. B.H. Flowers, Proc. R. Soc. London A 210, 497 (1952)

    Article  ADS  MATH  Google Scholar 

  4. D. Ruan, H.Z. Sun, Commun. Theor. Phys. 33, 221 (2000)

    MathSciNet  Google Scholar 

  5. I. Morrison, P.W. Pieruschka, B.G. Wybourne, J. Math. Phys. 32, 356 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. G. Dall'Agata, N. Prezas, J. High Energy Phys. 10, 103 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  7. B. de Carlos, A. Lukas, S. Morris, J. High Energy Phys. 12, 018 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  8. G. Ferretti, P. Salomonson, D. Tsimpis, J. High Energy Phys. 3, 004 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  9. A. Maas, S. Olejnik, J. High Energy Phys. 2, 070 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  10. C. Csaki, C. Grojean, H. Murayama, Phys. Rev. D 67, 085012 (2003)

    Article  ADS  Google Scholar 

  11. J. Patera, J. Math. Phys. 1, 3027 (1970)

    Article  MathSciNet  ADS  Google Scholar 

  12. R. Le Blanc, D.J. Rowe, J. Math. Phys. 29, 758 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. R. Le Blanc, D.J. Rowe, J. Math. Phys. 29, 767 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. A.M. Bincer, K. Riesselmann, J. Math. Phys. 34, 5935 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. R.G. Donnelly, S.J. Lewis, R. Pervine, Discret. Math. 306, 1285 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Borowiec, J. Lukierski, V. Lyakhovsky, M. Mozrzymas, V.N. Tolstoy, J. Math. Phys. 46, 103502 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  17. R. Coquereaux, R. Rais, E.H. Tahri, J. Math. Phys. 51, 092302 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  18. U.H. Niederer, L. O'Raifeartaigh, Fortschr. Phys. 22, 111 (1974)

    Article  MathSciNet  Google Scholar 

  19. E. Chacón, D. Levi, M. Moshinsky, J. Math. Phys. 17, 1919 (1976)

    Article  ADS  Google Scholar 

  20. M. Flato, C. Fronsdal, J. Math. Phys. 22, 1100 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  21. M. Flato, C. Fronsdal, Phys. Lett. B 97, 236 (1980)

    Article  ADS  Google Scholar 

  22. M. Lesimple, J. Math. Phys. 39, 6384 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. K.B. Alkalaev, M. Grigoriev, I.Yu. Tipunin, Nucl. Phys. B 823, 509 (2009)

    Article  ADS  MATH  Google Scholar 

  24. Q.Z. Han, H.Z. Sun, M. Zhang, D.H. Feng, J. Math. Phys. 26, 1822 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. D.P. Zhelobenko, Dokl. Akad. Nauk SSSR, 126, 635 (1959)

    Google Scholar 

  26. I.M. Gel'fand, V.A. Ponomarev, Russ. Math. Surveys 23, 1 (1968) (translation from Usp. Mat. Nauk 23

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. B. Gruber, W.C. Henneberger, Nuovo Cimento B 77, 203 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  28. A. Douglas, J. Math. Phys. 26, 1822 (2006)

    Google Scholar 

  29. H.C. Fu, J. Math. Phys. 32, 767 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. V. Chari, A. Pressley, A Guide to Quantum Groups (Cambridge University Press, Cambridge, 1994)

  31. J. Xiao, Canad. J. Math. 49, 771 (1997)

    Google Scholar 

  32. P. Casati, S. Minniti, V. Salari, J. Math. Phys. 51, 033515 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  33. B. Gruber, A.U. Klimyk, J. Math. Phys. 19, 2009 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. B. Gruber, A.U. Klimyk, J. Math. Phys. 25, 755 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. B. Gruber, A.U. Klimyk, Y.F. Smirnov, Nuovo Cimento 69, 97 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  36. B. Gruber, R. Lenczewski, J. Phys. A 16, 3703 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. R. Lenczewski, B. Gruber, J. Phys. A 19, 1 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. D. Ruan, L.H. Chen, W. Ruan, J. Math. Phys. 41, 7839 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. D. Ruan, Y.F. Jia, W. Ruan, J. Math. Phys. 42, 2718 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. H.D. Doebner, B. Gruber, M. Lorente, J. Math. Phys. 30, 594 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. H.C. Fu, C.P. Sun, J. Math. Phys. 31, 287 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. D. Ruan, C. Wu, H.Z. Sun, Commun. Theor. Phys. 40, 73 (2003)

    MathSciNet  MATH  Google Scholar 

  43. M. Moshinsky, J. Math. Phys. 4, 1128 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. C. Quesne, J. Math. Phys. 14, 366 (1973)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. H. Bacry, M. Boon, J. Math. Phys. 28, 2639 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. A. Klein, E.R. Marshalek, Rev. Mod. Phys. 63, 375 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  47. O. Civitarese, M. Reboiro, Phys. Rev. C 57, 3062 (1998)

    Article  ADS  Google Scholar 

  48. O. Civitarese, H.B. Geyer, M. Reboiro, Phys. Rev. C 73, 034306 (2006)

    Article  ADS  Google Scholar 

  49. O. Civitarese, P.O. Hess, J.G. Hirsch, M. Reboiro, Phys. Rev. C 61, 064303 (2000)

    Article  ADS  Google Scholar 

  50. T.D. Palevy, J. Phys. A: Math. Gen. 30, 8273 (1997)

    Article  ADS  Google Scholar 

  51. Y.Z. Zhang, M.D. Gould, J. Math. Phys. 46, 013505 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  52. T.D. Palevy, J. Phys. A: Math. Gen. 31, 5145 (1998)

    Article  ADS  Google Scholar 

  53. R. Dundarer, F. Gürsey, J. Math. Phys. 25, 431 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  54. Z. Yu, J. Phys. A: Math. Gen. 23, 939 (1990)

    Article  ADS  Google Scholar 

  55. R. Anishetty, M. Mathur, I. Raychowdhury, J. Math. Phys. 50, 053503 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  56. M. Mathur, I. Raychowdhury, R. Anishetty, J. Math. Phys. 51, 093504 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  57. D.M. Fradkin, J. Phys. A: Math. Gen. 27, 1261 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. R.A. Tello-Llanos, J. Phys. A: Math. Gen. 35, 3343 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. B.G. Wybourne, Classical Groups for Physicists (John Wiley & Sons, New York, 1974)

  60. J.E. Humphreys, Introductions to Lie Algebras and Representation Theory (Springer-Verlag, New York, 1972)

  61. D.N. Verma, Bull. Am. Math. Soc. 74, 160 (1968)

    Article  MATH  Google Scholar 

  62. M. Lorente, B. Gruber, J. Math. Phys. 25, 1674 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  63. I.N. Bemshtein, I.M. Gel'fand, S.I. Gel'fand, Funkcional. Anal. iPrilozen. 5, 1 (1971) (in Russian)

    Article  Google Scholar 

  64. L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Algebras and Superalgebras (Academic Press, San Diego, 2000)

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Tsinghua University, 100084, Beijing, P.R. China

    Hua-Jun Huang, You-Ning Li & Dong Ruan

Authors
  1. Hua-Jun Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. You-Ning Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dong Ruan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dong Ruan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, HJ., Li, YN. & Ruan, D. Indecomposable representations and oscillator realizations of the exceptional Lie algebra G2 . Eur. Phys. J. Plus 128, 66 (2013). https://doi.org/10.1140/epjp/i2013-13066-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2013-13066-4

Keywords

Search

Navigation