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 21( 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.
This paper is a first attempt to analyze in detail the number and structure of nontrivial polynomial conserved densities for a nonlinear evolution equation ut=P, P an arbitrary polynomial in the spatial derivatives of u. Our attention is here focused on the even order case, where stronger conclusions can be derived. Several criteria for nonexistence of conserved densities are afforded. The coexistence of conserved densities is shown to severely restrict both the evolution equation and the functional structure of such densities. Finally for the case of second order equations the problem is completely worked out.
An upper bound on the number of algebraically independent invariants in an enveloping algebra 𝒰 under the action of a Lie algebra G0 of derivations is obtained. We are able to determine the exact number of invariants for the case [G0,G0]=G0. This generalizes previous results about Casimir invariants.
A general discussion of the conservation laws for simple linear evolution systems is presented. The analysis is based upon an extension of the GeΓfand-Dikii symbolic algorithm to cover pseudo-differential operators. These techniques are applied to obtain all the conserved densities ρ\u] for the free Klein-Gordon and Dirac equations with nonzero mass.
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