Construction of the Invariants of the Simple Lie Groups

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 O₃, 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))