Communications in Mathematical Physics

, Volume 117, Issue 3, pp 451–500

Isospectral hamiltonian flows in finite and infinite dimensions

I. Generalized Moser systems and moment maps into loop algebras
  • M. R. Adams
  • J. Harnad
  • E. Previato

DOI: 10.1007/BF01223376

Cite this article as:
Adams, M.R., Harnad, J. & Previato, E. Commun.Math. Phys. (1988) 117: 451. doi:10.1007/BF01223376


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 curveXr 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.

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • M. R. Adams
    • 1
  • J. Harnad
    • 2
  • E. Previato
    • 3
  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA
  2. 2.Département de Mathémathiques Appliquées, Ecole PolytechniqueMontréalCanada
  3. 3.Department of MathematicsBoston UniversityBostonUSA