Poisson-Nijenhuis structures
Annales de l'I.H.P. Physique théorique, Volume 53 (1990) no. 1, p. 35-81
@article{AIHPA_1990__53_1_35_0,
     author = {Kosmann-Schwarzbach, Yvette and Magri, Franco},
     title = {Poisson-Nijenhuis structures},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     publisher = {Gauthier-Villars},
     volume = {53},
     number = {1},
     year = {1990},
     pages = {35-81},
     zbl = {0707.58048},
     mrnumber = {1077465},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1990__53_1_35_0}
}
Kosmann-Schwarzbach, Yvette; Magri, Franco. Poisson-Nijenhuis structures. Annales de l'I.H.P. Physique théorique, Volume 53 (1990) no. 1, pp. 35-81. http://www.numdam.org/item/AIHPA_1990__53_1_35_0/

[1] M. Adler and P. Van Moerbeke, Kowalewski's asymptotic method, Kac-Moody Lie algebras and regularization, Comm. Math. Phys., T. 83, 1982, pp. 83-106. | MR 648360 | Zbl 0491.58017

[2] R. Aminou, Groupes de Lie-Poisson et bigèbres de Lie, Thèse, Université des Sciences et Techniques de Lille Flandres Artois, June 1988.

[3] K.H. Bhaskara and K. Viswanath, Calculus on Poisson manifolds, Bull. London Math. Soc., T. 20, 1988, pp. 68-72. | MR 916078 | Zbl 0611.58002

[4] K. H, BHASKARA and K. Viswanath, Poisson algebras and Poisson manifolds, Pitman Research Notes in Math., Longman, 1988. | MR 960879 | Zbl 0671.58001

[5] R. Bkouche, Structures (K, A)-linéaires, C. R. Acad. Sci. Paris, T. 262, série A, 1966, pp. 373-376. | MR 197492 | Zbl 0139.25801

[6] J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geometry, T. 28, 1988, pp. 93-114. | MR 950556 | Zbl 0634.58029

[7] C. Buttin, Les dérivations des champs de tenseurs et l'invariant différentiel de Schouten, C. R. Acad. Sci. Paris, T. 269, série A, 1969, pp. 87-89. | MR 248662 | Zbl 0187.43904

[8] E. Caccese, On some involution theorems on twofold Poisson manifolds, Lett. Math. Physics, T. 15, 1988, pp. 193-200. | MR 948352 | Zbl 0652.58029

[9] A. Coste, P. Dazord and A. Weinstein, Groupoïdes symplectiques, Publ. Dép. Math. Université de Lyon-I, 2/A, 1987. | Numdam | Zbl 0668.58017

[10] C.M. De Barros, Espaces infinitésimaux, Cahiers Topol. Géom. Diff., T. 7, 1965. | Zbl 0147.41001

[11] C.M. De Barros, Opérateurs infinitésimaux sur l'algèbre des formes différentielles extérieures, C. R. Acad. Sci. Paris, T. 261, Groupe 1, 1965, pp. 4594-4597. | MR 188926 | Zbl 0139.39803

[12] I. Ya. Dorfman, Deformations of Hamiltonian structures and integrable systems, in Nonlinear phenomena and turbulence, Gordon and Breach, 1984. | MR 824793

[13] H. Flaschka, The Toda lattice in the complex domain, in Algebraic Analysis, in honor of Sato, M. KASHIWARA and T. KAWAI Eds., Vol. 1, Academic Press, 1988. | MR 992451 | Zbl 0688.58015

[14] A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms, Part I. Derivations in the graded ring of differential forms, Indag. Math., T. 18, 1956, pp. 338- 350 and 351-359. | MR 82554 | Zbl 0079.37502

[15] I.M. Gel'Fand and I.Ya. Dorfman, Hamiltonian operators and the classical Yang-Baxter equation, Funct. Anal. Appl., T. 16, No. 4. 1982, pp. 241-248. | MR 684122 | Zbl 0527.58018

[16] J.C. Herz, Pseudo-algèbres de Lie, C. R. Acad. Sci. Paris, T. 236, 1953, I, pp. 1935- 1937 and II, pp. 2289-2291. | Zbl 0050.03201

[17] H.G. Heuser, Functional Analysis, Wiley, New York 1982. | Zbl 0465.47001

[18] J. Huebschmann, Poison cohomology and quantization, J. für die Reine und Angew. Math. (to appear). | MR 1058984 | Zbl 0699.53037

[19] M.V. Karasev, Analogues of the objects of Lie group theory for nonlinear Poisson brackets, Math. U.S.S.R. Izvestiya, T. 28, No. 3, 1987, pp. 497-527 (in Russian, Izvestiya, T. 50, 1986). | MR 854594 | Zbl 0624.58007

[20] D. Kastler and R. Stora, Lie-Cartan pairs, J. Geom. and Physics, T. 2, No. 3, 1985. | MR 851120 | Zbl 0593.17009

[21] Y. Kosmann-Schwarzbach and F. Magri, Poisson-Lie groups and complete integrability, Part I. Drinfeld bigebras, dual extensions and their canonical representations, Ann. Inst. Henri Poincaré, série A (Physique théorique), T. 49, No. 4, 1988, pp. 433- 460, Parts II and III in preparation. | Numdam | MR 988946 | Zbl 0667.16005

[21 a] Y. Kosmann-Schwarzbach, The modified Yang-Baxter equation and bihamiltonian structures, in Differential Geometric Methods in Theoretical Physics (Chester, August 1988), A. SOLOMON Ed., World Scientific, Singapore, 1989, pp. 12-25. | MR 1124411

[22] B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math., T. 34, 1979, pp. 195-338. | MR 550790 | Zbl 0433.22008

[23] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in Elie Cartan et les mathématiques d'aujourd'hui, Société Mathématique de France, Astérisque, hors série, 1985, pp. 257-271. | MR 837203 | Zbl 0615.58029

[24] I.S. Krasil'Shchik, Schouten bracket and canonical algebras, in Global Analysis III, Lect. Notes Math., No. 1334, 1988, pp. 79-110. | MR 964696 | Zbl 0661.53059

[25] J. Lehmann-Lejeune, Étude des formes différentielles liées à certaines G-structures, C. R. Acad. Sci. Paris, T. 260, Groupe 1, 1965, pp. 1838-1841. | MR 180943 | Zbl 0135.40503

[26] J. Lehmann-Lejeune, Intégrabilité des G-structures définies par une 1-forme 0-déformable à valeurs dans le fibré tangent, Ann. Inst. Fourier, Grenoble, T. 16, No. 2, 1966, pp. 329-387. | Numdam | MR 212720 | Zbl 0145.42103

[27] A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geom., T. 12, 1977, pp. 253-300. | MR 501133 | Zbl 0405.53024

[28] Jiang-Hua Lu and A. Weinstein, Poisson Lie groups, dressing transformations and Bruhat decompositions, J. Diff. Geometry, T. 31, 1990, pp. 501-526. | MR 1037412 | Zbl 0673.58018

[29] K. Mackenzie, Lie groupoids and Lie algebroids in Differential Geometry, London Math. Soc. Lect. Notes, No. 124, Cambridge University Press, 1987. | Zbl 0683.53029

[30] F. Magri and C. Morosi, A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno, S 19, 1984, University of Milan.

[31] F. Magri, C. Morosi and O. Pagnisco, Reduction techniques for infinite-dimensional Hamiltonian systems: Some ideas and applications, Comm. Math. Phys., T. 99, 1985, pp. 115-140. | MR 791643 | Zbl 0602.58017

[31 a] F. Magri, Geometry and soliton equations, in La Mécanique Analytique de Lagrange et son héritage, Collège de France, September 1988 (to appear). | MR 1362129

[32] Sh. Majid, Non-commutative-geometric Groups by a Bicrossproduct construction: Hopf algebras at the Planck scale, Ph. D. Thesis, Harvard University, August 1988.

[33] P. Michor, Remarks on the Schouten-Nijenhuis bracket, Rendiconti Circ. Mat. Palermo (2), Suppl. No. 16, 1987, pp. 207-215. | MR 946726 | Zbl 0646.53013

[34] E. Nelson, Tensor Analysis, Princeton University Press, 1967. | Zbl 0152.39001

[35] A. Nijenhuis and R.W. Richardson Jr., Deformations of Lie algebra structures, J. Math. and Mech., T. 17, No. 1, 1967, pp. 89-105. | MR 214636 | Zbl 0166.30202

[36] R. Ouzilou, Hamiltonian actions on Poisson manifolds, in Symplectic Geometry, A. CRUMEYROLLE and J. GRIFONE Eds., Research Notes in Math., No. 80, Pitman, 1983. | MR 712169 | Zbl 0514.58010

[37] R.S. Palais, The Cohomology of Lie rings, Proc. Symp. Pure Math., 3, Amer. Math. Soc., 1961, pp. 130-137. | MR 125867 | Zbl 0126.03404

[38] J. Pradines, Théorie de Lie pour les groupoides différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris, T. 264, série A, 1967, pp. 245-248. | MR 216409 | Zbl 0154.21704

[39] G.S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc., T. 108, 1963, p. 195-222. | MR 154906 | Zbl 0113.26204

[40] J.A. Schouten, On the differential operators of first order in tensor calculus, Conv. di Geom. Differen., 1953, Cremonese, Rome, 1954. | MR 63750 | Zbl 0059.15301 | Zbl 0052.38204

[41] M.A. Semenov-Tian-Shansky, What is a classical r-matrix?, Funct. Anal. Appl., T. 17, No. 4, 1983, pp. 259-272. | Zbl 0535.58031

[42] A. Weinstein, Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo, Sect. 1 A, Math., T. 36, 1988, pp. 163-167. | MR 931446 | Zbl 0653.58012

[43] H. Yoshida, Integrability of generalized Toda lattice systems and singularities in the complex t-plane, in Nonlinear Integrable Systems, Classical Theory and Quantum Theory, World Scientific, Singapore, 1983. | MR 725708 | Zbl 0566.70021