L. Abellanas

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.

Conserved densities for nonlinear evolution equations. I. Even order case

This paper extends to nonlinear evolution equations of odd order the analysis of existence and structure of the polynomial conserved densities. The results for low order densities are similar to the case of even order. The situation for densities with high order derivatives is now radically different. An asymptotic algorithm is presented for the search of such densities, which are shown to be quadratic in the highest derivatives. The very existence of just one high order conserved density is shown to severely restrict the evolution equation, and in the third order case it leads, with some minor additional hypothesis, to the KdV family.

Evolution equations with high order conservation laws

An asymptotic algorithm presented in a previous paper is applied to investigate the possible structures of evolution equations ut=uM+K(u,...,uM−1),  M=3,5, which could be compatible with the existence of a conserved density ρ0(u), depending only on u, and with the existence as well of conserved densities with arbitrarily high‐order derivatives. For M=3 it is shown that the Calogero–Degasperis–Fokas equation is essentially the only nonpolynomial equation of that type. For the case ut=D[u4+Q], with Q(u,...,u3) a polynomial, we find a very narrow class of admissible structures for Q, typified by the few particular examples known up to date. Actually, there is in this case an essentially unique structure, modulo a Miura transformation.

Quantization from the algebraic viewpoint

We use the Weyl quantization W in a general context valid for any finite‐dimensional Lie algebra G, to derive an explicit formula for (P1,P2) ≡W−1([W (P1),W (P2)]), P1,P2 polynomials. In the particular case of the Heisenberg Lie algebra, this formula reduces to the familiar Moyal bracket.

Invariants in enveloping algebras under the action of Lie algebras of derivations

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.

Lie symmetries versus integrability in evolution equations

Motivated by some recent results obtained concerning the invariance properties of a particular family of generalized KdV equations, we investigate the possible significance of classical Lie invariance groups as a test for complete integrability.

Quasi-Lagrangian systems

A useful criterion is given which permits us to convert partial differential systems within a wide class into Lagrangian form.

Polynomial translation modules and Casimir invariants

The authors give a positive answer to a question about the existence of any direct link between two apparently unrelated facts for a specific family of solvable Lie algebras: the structure of their modules of functions on one side, and the algebraic form of their Casimir invariants on the other.