Invariant polynomials on Lie algebras of inhomogeneous unitary and special orthogonal groups

Abstract

The ring of invariant polynomials for the ad joint action of a Lie group on its Lie algebra is described for the inhomogeneous unitary and special orthogonal groups. In particular a new proof is given for the fact that this ring for the inhomogeneous Lorentz group is generated by two algebraically independent homogeneous polynomials of degrees two and four. During a series of lectures given at the Indian Statistical Institute in 1965, V. S. Varadarajan proved the following result: The ring of invariant polynomials for the adjoint action of the inhomogeneous special Lorentz group is generated by two algebraically independent homogeneous polynomials of degrees two and four. In this paper a technique used in the above is employed to obtain another proof of this result. More generally the above ring for the inhomogeneous unitary and special orthogonal groups is essentially described. A method due to J. Rosen is used to obtain invariant polynomials for the inhomogeneous groups from those of the homogeneous groups of next higher dimension. The author wishes to thank Professor R. Ranga Rao of the University of Illinois for suggesting the problem and making available the above mentioned lecture notes. 1. We now present some definitions, notation and preliminary results. Let i7: G -GL(V) be a representation of a group G on a finite dimensional vector space V. Let S(V) denote the symmetric algebra of V and recall that S(V) = fl m=OS(V), where S (V) is the subspace of homogeneous polynomials of degree n. We shall let I(G, V, i7) denote the algebra of polynomials in S(V) invariant under the induced action of G on S(V); note that I(G, V, 7i) = =01S (v) r)n I(G, V, 77+). Now let V* be the dual space of V and let P(v) = Received by the editors July 1, 1971. AMS 1970 subject classifications. Primary 22E43, 22E45; Secondary 15A72.