Linearization of Hamiltonian systems, Jacobi varieties and representation theory

Abstract

The purpose of this paper is twofold: first we show that all the systems discussed in Adler and van Moerbeke [2] (paper I) in connection with KacMoody Lie algebras can be linearized according to a general scheme common to all of them reminiscent of Mumford and van Moerbeke’s treatment of the Toda lattice [16]. The methods reflect the decomposition of the Lie algebra as explained in I and therefore are divided into two different sections dealing on the one hand with Toda-type flows and on the other hand with flows of spinning top type (Sections 2 and 3). The second part of the paper adresses the following problem: each Lie algebra representation leads to a different curve and therefore a different Jacobi variety; therefore one might expect the linearization to depend on the representation; i.e. the same flow would lead to essentially different solution by quadratures, depending on the representation. We show this is not the case, because the Jacobi varieties corresponding to higher-order representations all contain one or several copies of the Jacobi variety going with the fundamental representation. To show this, use is made of the theory of correspondences: a correspondence is established between the fundamental curve and the curves corresponding to higher-dimensional representations; the latter curves are Galois extensions of the fundamental curve. Such a correspondence induces a homomorphism between the associated Jacobi varieties, which is shown to be different from zero and injective; the first statement follows from an inequality of Castelnuovo and the second from the irreducibility of the Jacobians for Toda-like