On an infinitesimal approach to semisimple Lie groups and raising and lowering operators of O(n) and U(n)

A purely algebraic approach to the evaluation of the fundamental Wigner coefficients and reduced matrix elements of O(n) and U(n) is given. The method employs the explicit use of projection operators which may be constructed using the polynomial identities satisfied by the infinitesimal generators of the group. As an application of this technique, a certain set of raising and lowering operators for O(n) and U(n) are constructed. They are simpler in appearance than those previously constructed since they may be written in a compact product form. They are, moreover, Hermitian conjugates of one another, and therefore are easily normalized.