Linearization of Hamiltonian systems, Jacobi varieties and representation theory. (English) Zbl 0455.58010

In the note under review the authors show that all systems discussed in their previous note [Adv. Math. 38, 267–317 (1980; Zbl 0455.58017)] 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, and prove the independence of the linearization on the representation. The paper also contains a beautiful appendix in which the authors sketch the main concepts and theorems of the theory of correspondence in a style reminiscent of the classic Italian geometers.


37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
53D50 Geometric quantization
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14E05 Rational and birational maps
14H40 Jacobians, Prym varieties


Zbl 0455.58017
Full Text: DOI


[1] Adler, M., On a trace functional for formal pseudo-differential operators and the symplectic structure of the \(KdV\) type equations, (Proceedings, Conf. on Nonlinear Analysis. Proceedings, Conf. on Nonlinear Analysis, Calgary, June 1978. Proceedings, Conf. on Nonlinear Analysis. Proceedings, Conf. on Nonlinear Analysis, Calgary, June 1978, Springer Lecture notes, vol. 755 (1980), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0393.35058
[2] Adler, M.; van Moerbeke, P., Completely integrable systems, Euclidean Lie algebras, and curves. Euclidean Lie algebras, and curves, Advances in Math., 36, 267-317 (1980) · Zbl 0455.58017
[3] Castelnuovo, G., Sulle Serie algebriche di gruppi di punti appartenenti and una curva algebrica, Rend. Accad. Lincei, 5, 15, 337-344 (1906)
[4] Chow, W. L., The Jacobean of an algebraic curve, Amer. J. Math., 76, 453-476 (1954) · Zbl 0056.14404
[5] Coolidge, J. L., A Treatise on Angebraic Plane Curves (1959), Dover: Dover New York · Zbl 0085.36403
[6] Russian Math. Surveys, 31, No. 1 (1976)
[7] Igusa, J. I., Fibre systems of Jacobian varieties, Amer. J. Math., 78, 171-199 (1956) · Zbl 0074.15803
[8] Lefschetz, S., A theorem on correspondences of algebraic curves, Trans. Amer. Math. Soc. (1928)
[9] Lefschetz, S., Correspondences between algebraic curves, Ann. of Math., 28, 342-354 (1926)
[10] McKean, H. P.; van Moerbeke, P., The spectrum of Hill’s equation, Invent. Math., 30, 217-276 (1973) · Zbl 0319.34024
[11] McKean, H. P.; van Moerbeke, P., About Hill and Toda curves, Comm. Pure Appl. Math., 23, 23-42 (1980) · Zbl 0422.14017
[12] Matsusaka, T., On the algebraic construction of the Picard variety, Japan J. Math., 22, 51-62 (1952) · Zbl 0049.22801
[13] Matsusaka, T., Specialization of cycles on a projective model, Mem. Coll. Sci. Univ. Kyoko, Ser. A, 26 (1951) · Zbl 0045.24201
[14] van Moerbeke, P., The spectrum of Jacobi matrices, Invent. Math., 37, 45-81 (1976) · Zbl 0361.15010
[15] van Moerbeke, P., About isospectral deformations of discrete Laplacians (dimensions 1 and 2), (Proceedings, Conf. on Non-Linear Analysis. Proceedings, Conf. on Non-Linear Analysis, Calgary, June 1978. Proceedings, Conf. on Non-Linear Analysis. Proceedings, Conf. on Non-Linear Analysis, Calgary, June 1978, Springer Lecture Notes, vol. 755 (1979), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0464.58019
[16] van Moerbeke, P.; Mumford, D., The spectrum of difference operators and algebraic curves, Acta Math., 143, 93-154 (1979) · Zbl 0502.58032
[17] Ratiu, T., Involution theorems, (Proceedings, Lowell Conference. Proceedings, Lowell Conference, Springer Lecture Notes (1980), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0435.58014
[18] \( \textsc{T. Ratiu}n\); \( \textsc{T. Ratiu}n\) · Zbl 0432.70011
[19] T. Ratiu; T. Ratiu
[20] T. Ratiu and P. van Moerbeke; T. Ratiu and P. van Moerbeke
[21] Shimura, G., Reduction of algebraic varieties with respect to a discrete valuation of the basic field, Amer. J. of Math., 77, 143-176 (1955) · Zbl 0065.36701
[22] Weil, A., Courbes algebriques et variétés abéliennes (1971), Hermann: Hermann Paris · Zbl 0208.49202
[23] Whittaker, Analytical Dynamics (1921), Oxford Univ. Press: Oxford Univ. Press London/New York
[24] Arnold, V. I., Mathematical Methods of Classical Mechanics (1978), Springer-Verlag: Springer-Verlag New York · Zbl 0386.70001
[25] Devaney, R., Transversal homoclinic orbits in an integrable system, Amer. J. Math., 100, 631-648 (1978) · Zbl 0406.58019
[26] Mattuck, A., Secant bundles on symmetric products, Amer. J. Math., 87, 4 (1965) · Zbl 0196.53503
[27] Reiman, A.; Semenov-Tian-Shonsky, M., Reduction of Hamiltonian Systems, Affine Lie algebras and Lax equations, Invent. Math., 54, 81-101 (1979) · Zbl 0403.58004
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