Kronecker webs, bihamiltonian structures, and the method of argument translation. (English) Zbl 0994.37034
This paper starts with an interesting review on some topics about integrability of bi-Hamiltonian systems. Then the author shows that manifolds which parametrize values of first integrals of finite-dimensional bi-Hamiltonian systems carry a geometric structure, called Kronecker web. There are two opposite functors between Kronecker webs and integrable bi-Hamiltonian strucures having the property that one is the left inverse of the other. It is also proven that if the bi-Hamiltonian strucure admits an appropriate anti-involution structure then the functors are in fact mutually inverse, for “small” open subsets. This implis the conjecture of I. M. Gelfand and I. Zakharevich [Sel. Math., New Ser. 6, 131-183 (2000; Zbl 0986.37060)] that on a dense open subset the bi-Hamiltonian structure of \(\mathfrak{g}^\star\) is flat if \(\mathfrak g\) is semisimple.
Reviewer: Paulo R.Rodrigues (Rio de Janeiro)
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 53A60 | Differential geometry of webs |
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
Citations:
Zbl 0986.37060References:
| [1] | J. F. Adams,Lectures on Lie Groups, W. A. Benjamin, Inc., New York, Amsterdam, 1969. Russian transl.: ??. ?????,?????? ?? ??????? ??, ?????, ?., 1979. |
| [2] | ?. ?. ???????, ? ????????????? ?????????? ????????? ?2 ??????????? ?????? ??, ????. ???, ???. ???.37 (1982), No. 2, 29-34. English transl.: L. V. Antonyan, On classification of homogeneous elements of ?2-graded semisimple Lie algebras, Mosc. Univ. Math. Bull.37 (1982), No. 2, 36-43. |
| [3] | ?. ?. ???????,??????????????? ?????? ???????????? ????????, ?????, ?., 1974. English transl.: V. I. Arnold,Mathematical Methods of Classical Mechanics, Graduate Texts in Math.60, Springer-Verlag, New York, Heidelberg, Berlin, 1978. |
| [4] | ?. ?. ???????, ?. ?. ?????????,??????????????? ?????????, ????. ?????. ??????????. ????. ???????., ?. 4, ???????????? ???????-4, ??????, ?., 1985, 5-139. English transl. V. I. Arnold, A. B. Givental,Symplectic geometry, in:Dynamical Systems IV.Symplectic Geometry and its Applications, Encycl. Math. Sci., Springer-Verlag, Berlin, Heidelberg, New York, 1989. · Zbl 0527.05011 |
| [5] | ?. ?. ????????,?????? ??????? ?????? ???????? ?? ???????? ?? ? ??????? ????? ???? ??????? ? ?????????, ???. ?? ????, ???. ???.55 (1991), No. 1, 68-92. English transl.: A. V. Bolsinov,Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Math. USSR, Izv.38 (1992), No. 1, 69-90. |
| [6] | N. Bourbaki,Groupes et Algèbres de Lie, Ch. VII, VIII, Hermann, Paris, 1975. Russian transl.: ?. ???????,?????? ? ??????? ??. ????? VII, VIII, ???, ?., 1978. |
| [7] | ?. ?. ?????????, ?. ?. ???????,??????????? ?????? ? ?????? ?????????, ?????, ?., 1986. English transl.: L. D. Faddeev, L. A. Takhtajan,Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, Berlin, 1987. · Zbl 0634.68122 |
| [8] | A. S. Fokas and B. Fuchssteiner,On the structure of symplectic operators and hereditary symmetries, Lett. Nuovo Cimento (2)28 (1980), No. 8, 299-303. · doi:10.1007/BF02798794 |
| [9] | ?. ?. ?????????,?????? ??????, ?????????, ?., 1954. English transl.: F. R. Gantmacher,The Theory of Matrices, Vols. 1, 2, Chelsea Publ. Co., New York, 1959. |
| [10] | ?. ?. ????????, ?. ?. ???????,???????????? ????????? ? ????????? ? ???? ?????????????? ?????????, ?????. ????. ? ??? ????.13 (1979), No. 4, 13-30. English transl.: I. M. Gelfand, I. Ya. Dorfman,Hamiltonian operators and algebraic structures associated with them, Funct. Anal. Appl.13 (1980), 248-262. |
| [11] | ?. ?. ????????, ?. ?. ?????????,???????????? ?????? ????? ?????????????????? ???????????????? ?????????? 3-?? ??????? ?? S 1, ?????. ????. ? ??? ????.23 (1989), No. 2, 1-11. English transl.: I. M. Gelfand, I. S. Zakharevich,Spectral theory for a pencil of skew-symmetrical differential operators of third order on S 1, Func. Anal. Appl.23 (1989), No. 2, 85-93. |
| [12] | ?,Webs, Veronese curves, and bihamiltonian systems, J. of Func. Anal.99 (1991), 150-178. · Zbl 0739.58021 · doi:10.1016/0022-1236(91)90057-C |
| [13] | ?,On the local geometry of bihamiltonian structures, in:The Gelfand Mathematical Seminar, 1990-1992, Birkhäuser, Boston, 1993, pp. 51-112. |
| [14] | ?,The spectral theory for a pencil of skew symmetrical differential operators of the third order, Comm. Pure Appl. Math.47 (1994), No. 8, 1031-1041. · Zbl 0815.47009 · doi:10.1002/cpa.3160470802 |
| [15] | ?,Webs, Lenard schemes, and the local geometry of bihamiltonian Toda and Lax structures, Sel. Math., New ser.6 (2000), No. 2, 131-183. · Zbl 0986.37060 · doi:10.1007/PL00001387 |
| [16] | V. Guillemin, S. Sternberg,Geometric Asymptotics, Mathematical Surveys, No. 14, Amer. Math. Soc., Providence, R.I., 1977. Russian transl.: ?. ???????, ?. ?????????,?????????????? ???????????, ???, ?., 1981. |
| [17] | ?. ?. ???????,??????????? ???? ??????????, ????? ???. ?? ????. ? ????. ???????8 (1950), 355-363. (G. B. Gurevi?,Canonization of a pair of bivectors, Trudy seminara po vectornomu i tenzornomu analizu8 (1950), 355-363. (Russian).) · Zbl 0041.48406 |
| [18] | ?. ?. ????????,????????? ??????? ??, ???31 (1976), No. 4(190), 57-76. English transl.: A. A. Kirillov,Local Lie algebras, Rus. Math. Surveys31 (1976), No. 4, 55-75. |
| [19] | Y. Kosmann-Schwarzbach, F. Magri,Lax-Nijenhuis operators for integrable systems, J. Math. Phys.37 (1996), No. 12, 6173-6197. · Zbl 0862.58034 · doi:10.1063/1.531771 |
| [20] | B. Kostant,The solution to a generalized Toda lattice and representation theory, Adv. in Math.34 (1979), No. 3, 195-338. · Zbl 0433.22008 · doi:10.1016/0001-8708(79)90057-4 |
| [21] | P. D. Lax,Almost periodic solutions of the KdV equation, SIAM Rev.18 (1976), No. 3, 351-375. · Zbl 0329.35015 · doi:10.1137/1018074 |
| [22] | F. Magri,A simple model of the integrable Hamiltonian equation, J. Math. Physics19 (1978), No. 5, 1156-1162. · Zbl 0383.35065 · doi:10.1063/1.523777 |
| [23] | F. Magri,On the geometry of soliton equations, preprint, 1988. · Zbl 0781.35059 |
| [24] | F. Magri,On the geometry of soliton equations. Geometric and algebraic structures in differential equations, Acta Appl. Math.41 (1995), No. 1-3, 247-270. · Zbl 0842.58048 · doi:10.1007/BF00996115 |
| [25] | H. McKean, private communication, 1990. |
| [26] | ?. ?. ???????, ?. ?. ???????,????????? ?????? ?? ????????????? ?????? ??, ???. ?? CCCP, ???. ???.42 (1978), No. 2, 396-415. English transl.: A. S. Mi??enko, A. T. Fomenko,Euler equation on finite-dimensional Lie groups, Math USSR, Izv.12 (1978), 371-389. |
| [27] | C. Morosi, L. Pizzocchero,On the Euler equation: bi-Hamiltonian structure and integrals in involution, Lett. Math. Phys.37 (1996), No. 2, 117-135. · Zbl 0852.58050 · doi:10.1007/BF00416015 |
| [28] | A. Panasyuk,Symplectic realizations of bihamiltonian structures, preprint, 1998. |
| [29] | A. Panasyuk,Veronese webs for bi-Hamiltonian structures of higher corank, in:Poisson Geometry (Warsaw, 1998), Polish Acad. Sci., Warsaw, 2000, pp. 251-261. · Zbl 0996.53053 |
| [30] | M.-H. Rigal,Géométrie globale des systèmes bihamiltoniens réguliers de rang maximum en dimension 5, C. R. Acad. Sci. Paris Sér. I Math.321 (1995), No. 11, 1479-1484. · Zbl 0853.58055 |
| [31] | M.-H. Rigal,Tissus de Véronèse en dimension 3, in:Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1994-1995 (Montpellier), Univ. Montpellier II, Montpellier, 1995, pp. iv, 63-68. |
| [32] | M.-H. Rigal,Systèmes bihamiltoniens en dimension impaire, Ann. Sci. École Norm. Sup. (4)31 (1998), No. 3, 345-359. · Zbl 0943.37032 |
| [33] | J. Sekiguchi, Y. Shimizu,Simple singularities and infinitesimally symmetric spaces, Proc. Jap. Acad. Ser. A Math. Sci.57, (1981), No. 1, 42-46. · Zbl 0481.58009 · doi:10.3792/pjaa.57.42 |
| [34] | S. Sternberg,Lectures on Differential Geometry, second ed., Chelsea Publ. Co., New York, 1983. · Zbl 0518.53001 |
| [35] | R. C. Thompson,Pencils of complex and real symmetric and skew matrices, Lin. Alg. Appl.147 (1991), 323-371. · Zbl 0726.15007 · doi:10.1016/0024-3795(91)90238-R |
| [36] | F.-J. Turiel,Classification locale d’un couple de formes symplectiques Poisson-compatibles, C. R. Acad. Sci. Paris Sér. I Math.308 (1989), No. 20, 575-578. · Zbl 0673.53022 |
| [37] | F.-J. Turiel,C ?-équivalence entre tissus de Veronese et structures bihamiltoniennes, C. R. Acad. Sci. Paris Sér. I Math.328 (1999), No. 10, 891-894. · Zbl 0929.53015 |
| [38] | F.-J. Turiel,Mémoire â l’appui du projet de Note: C ?-équivalence entre tissus de Veronese et structures bihamiltoniennes, preprint, 1999. |
| [39] | F.-J. Turiel,Mémoire â l’appui du projet de Note: Tissus de Veronese analytiques de codimension supérieure et structures bihamiltoniennes, preprint, 2000. |
| [40] | F.-J. Turiel,Tissus de Veronese analytiques de codimension supérieure et structures bihamiltoniennes, C. R. Acad. Sci. Paris Sér. I Math.331 (2000), No. 1, 61-64. · Zbl 0969.53005 |
| [41] | H. W. Turnbull, A. C. Aitken,An Introduction to the Theory of Canonical Matrices, Dover Publ. Inc., New York, 1961. · Zbl 0096.00801 |
| [42] | A. Weinstein,The local structure of Poisson manifolds, J. Diff. Geom.18 (1983), No. 3, 523-557. · Zbl 0524.58011 |
| [43] | I. Zakharevich,Kronecker webs, bihamiltonian structures, and the method of argument translation, archived as math.SG/9908034. · Zbl 0994.37034 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
