Vector Operators and a Polynomial Identity for SO(n)

Abstract

It is shown that if α denotes an n × n antisymmetric matrix of operators αpq,p,q = 1, 2, …, n, which satisfy the commutation relations characteristic of the Lie algebra of SO(n), then α satisfies an nth degree polynomial identity. A method is presented for determining the form of this polynomial for any value of n. An indication is given of the simple significance of this identity with regard to the problem of resolving an arbitrary n-vector operator into n components, each of which is a vector shift operator for the invariants of the SO(n) Lie algebra.