Dual isomonodromic deformations and moment maps to loop algebras

@article{Harnad1993DualID,
  title={Dual isomonodromic deformations and moment maps to loop algebras},
  author={J. Harnad},
  journal={Communications in Mathematical Physics},
  year={1993},
  volume={166},
  pages={337-365}
}
  • J. Harnad
  • Published 18 January 1993
  • Mathematics
  • Communications in Mathematical Physics
The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those… 
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References

SHOWING 1-10 OF 20 REFERENCES

Darboux coordinates and Liouville-Arnold integration in loop algebras

AbstractDarboux coordinates are constructed on rational coadjoint orbits of the positive frequency part $$\tilde{\mathfrak{g}}^+$$ of loop algebras. These are given by the values of the spectral…

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function

Isospectral Hamiltonian flows in finite and infinite dimensions

AbstractThe approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional…

Dual moment maps into loop algebras

  • M. AdamsJ. HarnadJ. Hurtubise
  • Mathematics
  • 1990
AbstractMoment maps are defined from the space of rank-r deformations of a fixedn xn matrixA to the duals $$(\widetilde{g1}(r)^ + )^* , (\widetilde{g1}(n)^ + )^* $$ of the positive half of the loop…

Hamiltonian Structure of Equations Appearing in Random Matrices

The level spacing distributions in the Gaussian Unitary Ensemble, both in the “bulk of the spectrum,” given by the Fredholm determinant of the operator with the sine kernel \(\frac{{\sin \;\pi (x -…

Differential Equations for Quantum Correlation Functions

The quantum nonlinear Schrodinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation…

Isospectral hamiltonian flows in finite and infinite dimensions

AbstractA 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)}^ +…

Level-spacing distributions and the Airy kernel

Introduction to Random Matrices

Here I = S j (a2j 1,a2j) andI(y) is the characteristic function of the set I. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in I is equal to �(a). Also �(a) is a…