Construction of the Invariants of the Simple Lie Groups

  • L. O’Raifeartaigh

Abstract

To introduce the problem which I should like to consider in this talk, let me first make some remarks about the three-dimensional rotation group O 3, with which we are familiar. For this group, the infinitesimal generators T i, i = 1, 2, 3, satisfy the commutation relations
$${\text{[}}{T_i}{T_j}{\text{] = }}i{T_k}{\text{ (}}i,j,k{\text{ cyclic)}}$$
(1)
and there exists a polynomial in the T i, namely,
$$\mathop T\nolimits^2 = \;\mathop T\nolimits_1^2 + \mathop T\nolimits_2^2 + \mathop T\nolimits_3^2$$
(2)
which has the property that it commutes with all of the infinitesimal generators:
$$[{T^2},{T_i}]\; = \;0\;\;(i = 1,2,3)$$
(3)

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References

  1. 1.
    G. Racah, “Group Theory and Spectroscopy,” CERN, Reprint 61-8 (1961).Google Scholar
  2. 2.
    G. Racah, Rend. Lincei 8: 108 (1950).MathSciNetMATHGoogle Scholar
  3. 3.
    B. Gruber and L. O’Raifeartaigh, J. Math. Phys. 5: 1796 (1964).MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    S. Okubo, Progr. Theoret. Phys. 27: 949 (1962).ADSMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press 1966

Authors and Affiliations

  • L. O’Raifeartaigh
    • 1
  1. 1.Dublin Institute for Advanced StudiesDublinIreland

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