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Isospectral hamiltonian flows in finite and infinite dimensions

I. Generalized Moser systems and moment maps into loop algebras

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, 30602, Athens, GA, USA

    M. R. Adams

  2. Département de Mathémathiques Appliquées, Ecole Polytechnique, Succ. “A”, C.P. 6079, H3C 3A7, Montréal, Qué, Canada

    J. Harnad

  3. Department of Mathematics, Boston University, 02215, Boston, MA, USA

    E. Previato

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  1. M. R. Adams
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  2. J. Harnad
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  3. E. Previato
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Communicated by A. Jaffe

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

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Adams, M.R., Harnad, J. & Previato, E. Isospectral hamiltonian flows in finite and infinite dimensions. Commun.Math. Phys. 117, 451–500 (1988). https://doi.org/10.1007/BF01223376

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