, Volume 117, Issue 3, pp 451-500

Isospectral hamiltonian flows in finite and infinite dimensions

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

Abstract

A moment map \(\tilde J_r :M_A \to (\widetilde{gl(r)}^ + )^*\) is constructed from the Poisson manifold ℳA of rank-r perturbations of a fixedN×N matrixA to the dual \((\widetilde{gl(r)}^ + )^*\) of the positive part of the formal loop algebra \(\widetilde{gl(r)}\) =gl(r)⊗ℂ[[λ, λ−1]]. The Adler-Kostant-Symes theorem is used to give hamiltonians which generate commutative isospectral flows on \((\widetilde{gl(r)}^ + )^*\) . The pull-back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in ℳA. The latter may be identified with flows on finite dimensional coadjoint orbits in \((\widetilde{gl(r)}^ + )^*\) and linearized on the Jacobi variety of an invariant spectral curveX r which, generically, is anr-sheeted Riemann surface. Reductions of ℳA are derived, corresponding to subalgebras ofgl(r, ℂ) andsl(r, ℂ), determined as the fixed point set of automorphism groupes generated by involutions (i.e., all the classical algebras), as well as reductions to twisted subalgebras of \(\widetilde{sl(r,\mathbb{C}})\) . The theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of PDE's.

This research was partially supported by NSF grants MCS-8108814 (A03), DMS-8604189, and DMS-8601995
Communicated by A. Jaffe