A general setting for Casimir invariants

This paper contains a general description of the theory of invariants under the adjoint action of a given finite-dimensional complex Lie algebra G, with special emphasis on polynomial and rational invariants. The familiar ’’Casimir’’ invariants are identified with the polynomial invariants in the enveloping algebra 𝒰 (G). More general structures (quotient fields) are required in order to investigate rational invariants. Some useful criteria for G having only polynomial or rational invariants are given. Moreover, in most of the physically relevant Lie algebras the exact computation of the maximal number of algebraically independent invariants turns out to be very easy. It reduces to finding the rank of a finite matrix. We apply the general method to some typical examples.