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Banach Poisson–Lie Groups and Bruhat–Poisson Structure of the Restricted Grassmannian

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The first part of this paper is devoted to the theory of Poisson–Lie groups in the Banach setting. Our starting point is the straightforward adaptation of the notion of Manin triples to the Banach context. The difference with the finite-dimensional case lies in the fact that a duality pairing between two non-reflexive Banach spaces is necessary weak (as opposed to a strong pairing where one Banach space can be identified with the dual space of the other). The notion of generalized Banach Poisson manifolds introduced in this paper is compatible with weak duality pairings between the tangent space and a subspace of the dual. We investigate related notion like Banach Lie bialgebras and Banach Poisson–Lie groups, suitably generalized to the non-reflexive Banach context. The second part of the paper is devoted to the treatment of particular examples of Banach Poisson–Lie groups related to the restricted Grassmannian and the KdV hierarchy. More precisely, we construct a Banach Poisson–Lie group structure on the unitary restricted Banach Lie group which acts transitively on the restricted Grassmannian. A“dual” Banach Lie group consisting of (a class of) upper triangular bounded operators admits also a Banach Poisson–Lie group structure of the same kind. We show that the restricted Grassmannian inherits a generalized Banach Poisson structure from the unitary Banach Lie group, called Bruhat–Poisson structure. Moreover the action of the triangular Banach Poisson–Lie group on it is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian.

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References

  1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications (second edition), Applied Mathematical Sciences 75. Springer, New York (1988)

    Google Scholar 

  2. Arazy, J.: Some remarks on interpolation theorems and the boundness of the triangular projection in unitary matrix spaces. Integral Equ. Oper. Theory 1(4), 453–495 (1978)

    Google Scholar 

  3. Balleier, C., Wurzbacher, T.: On the geometry and quantization of symplectic Howe pairs. Mathematische Zeitschrift 271(1), 577–591 (2012)

    Google Scholar 

  4. Beltiţă, D.: Iwasawa decompositions of some infinite-dimensional Lie groups. Trans. Am. Math. Soc. 361(12), 6613–6644 (2009)

    Google Scholar 

  5. Beltiţă, D.: Functional analytic background for a theory of infinite-dimensional reductive Lie groups, Developments and trends in infinite-dimensional Lie theory, 367–392, Progr. Math., 288, Birkhäuser Boston, Inc., Boston, MA, (2011)

  6. Beltiţă, D., Golinski, T., Tumpach, A.B.: Queer Poisson brackets. J. Geom. Phys. 132, 358–362 (2018)

    Google Scholar 

  7. Beltiţă, D., Neeb, K.-H.: Geometric characterization of Hermitian algebras with continuous inversion. Bull. Aust. Math. Soc. 81(1), 96–113 (2010)

    Google Scholar 

  8. Beltiţă, D., Ratiu, T.S., Tumpach, A.B.: The restricted Grassmannian, Banach Lie-Poisson spaces and coadjoint orbits. J. Funct. Anal. 247(1), 138–168 (2007)

    Google Scholar 

  9. Borcherds, R., Haiman, M., Reshetikhin, N., Serganova, V.: Berkeley Lectures on Lie Groups and Quantum Groups, Geraschenko, A., Johnson-Freyd, T., (eds.) http://math.berkeley.edu/~theojf/LieQuantumGroups.pdf

  10. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010)

    Google Scholar 

  11. Cabau, P., Pelletier, F.: Almost Lie structures on an anchored Banach bundle. J. Geom. Phys. 62(11), 2147–2169 (2012)

    Google Scholar 

  12. Davidson, K.R.: Nest algebras, Pitman Research Notes in Mathematics Series, vol. 191. Longman Scientific and Technical Pub. Co., New York (1988)

    Google Scholar 

  13. de Bièvre, S., Genoud, F., Rota Nodari, S.: Orbital stability: Analysis meets Geometry, Nonlinear optical and atomic systems, 147–273, Lecture Notes in Math., 2146, CEMPI Ser., Springer, Cham, (2015)

  14. Dobrogowska, A., Odzijewicz, A.: Integrable Hamiltonian systems related to the Hilbert–Schmidt ideal. J. Geom. Phys. 61(8), 1426–1445 (2011)

    Google Scholar 

  15. Drinfel’d, V.G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang–Baxter equations. Soviet Math. Dokl. 27(1), 68–71 (1983)

    Google Scholar 

  16. Gay-Balmaz, F., Vizman, C.: Dual pairs in fluid dynamics. Ann. Glob. Anal. Geom. 41(1), 1–24 (2012)

    Google Scholar 

  17. Gay-Balmaz, F., Vizman, C.: A dual pair for free boundary fluids. Int. J. Geom. Methods Mod. Phys. 12(7), (2015)

  18. Gay-Balmaz, F., Vizman, C.: Dual pairs for non-abelian fluids, Geometry, mechanics, and dynamics, Fields Inst. Commun. 73, 107–135, Springer, New-York (2015)

  19. Gohberg, I.C., Krein, M.G.: Theory and Applications of Volterra operators on Hilbert spaces, Translations of Mathematical Monographs (2nd ed.), vol. 24, Am. Math, Soc, Providence, RI (1970)

  20. Goliński, T., Odzijewicz, A.: Hierarchy of Hamiltonian equations on Banach Lie–Poisson spaces related to restricted Grassmannian. J. Funct. Anal. 258, 3266–3294 (2010)

    Google Scholar 

  21. Khesin, B., Zakharevich, I.: Poisson–Lie group of pseudodifferential symbols. Commun. Math. Phys. 171(3), 475–530 (1995)

    Google Scholar 

  22. Kirillov, A.A.: Merits and Demerits of the orbit method, Bulletin (New Series) of the American Mathematical Society, 36(4), 433–488

  23. Kosmann-Schwarzbach, Y., Magri, F.: Poisson–Lie groups and complete integrability. Ann. Inst. Henri Poincaré 49(4), 433–460 (1988)

    Google Scholar 

  24. Kosmann-Schwarzbach, Y.: Poisson–Lie groups and beyond. J. Math. Sci. 82(6), 3807–3813 (1996)

    Google Scholar 

  25. Kosmann-Schwarzbach, Y.: Lie bialgebras, Poisson Lie groups and dressing transformations, Integrability of Nonlinear Systems, Second edition, Lectures Notes in Physics 638, Springer, pp. 107–173 (2004)

  26. Kosmann-Schwarzbach, Y., Magri, F.: Lax–Nijenhius operators for integrable systems. J. Math. Phys. 37(12), 6173–6197 (1996)

    Google Scholar 

  27. Kriegl, A., Michor, P.W.: The convenient setting of global analysis, Mathematical surveys and Monographs. Am. Math. Soc. 53, (1997)

  28. Kwapien, S., Pelczynski, A.: The main triangle projection in matrix spaces and its applications. Studia Math. 34, 43–68 (1970)

    Google Scholar 

  29. Lang, S.: Differential and Riemannian Manifolds, Graduate Texts in Mathematics, vol. 160. Springer, New York (1995)

    Google Scholar 

  30. Lang, S.: Fundamentals of Differential Geometry (corrected second printing), Graduate Texts in Mathematics, vol. 191. Springer, New York (2001)

    Google Scholar 

  31. Lu, J.-H.: Multiplicative and Affine Poisson Structures on Lie groups, Ph.D. thesis, UC Berkeley (1991)

  32. Lu, J.-H., Weinstein, A.: Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Differ. Geom. 31(2), 501–526 (1990)

    Google Scholar 

  33. Macaev, V.I.: Volterra operators produced by perturbation of self adjoint operators. Soviet Math. Dokl. 2, 1013–1016 (1961)

    Google Scholar 

  34. Marle, C.-M.: Introduction aux groupes de Lie-Poisson. In: Alves, A.S., Craveiro de Carvalho, F.J., Pereira da Silva, J.A. (eds.) Geometria, Fisica-Matemática e Outros Ensaios, pp. 149–170. Departamento de Matemática, Universidad de Coimbra, Coimbra (1998)

    Google Scholar 

  35. Neeb, K.-H., Sahlmann, H., Thiemann, T.: Weak Poisson structures on infinite dimensional manifolds and Hamiltonian actions, Lie Theory and Its Applications in Physics, Varna, Bulgaria, June 2013, Springer Proceedings in Mathematics and Statistics, Vol. 111, pp. 105–135, Springer, Tokyo (2014)

  36. Neeb, K.-H.: Towards a Lie theory of locally convex groups. Jpn. J. Math. 1(2), 291–468 (2006)

    Google Scholar 

  37. Odzijewicz, A., Ratiu, T.S.: Banach Lie–Poisson spaces and reduction. Commun. Math. Phys. 243(1), 1–54 (2003)

    Google Scholar 

  38. Odzijewicz, A., Ratiu, T.S.: Induction for weak symplectic Banach manifolds. J. Geom. Phys. 58(6), 701–719 (2008)

    Google Scholar 

  39. Odzijewicz, A., Ratiu, T.S.: Induced and coinduced Lie–Poisson spaces and integrability. J. Funct. Anal. 255(5), 1225–1272 (2008)

    Google Scholar 

  40. Pelletier, F.: Integrability of weak distributions on Banach manifolds. Indag. Math. (N.S.) 23(3), 214–242 (2012)

    Google Scholar 

  41. Pressley, A.: Loop Groups, Grassmannians and KdV equations, Infinite-dimensional groups with applications. Publ. Math. Sci. Res. Inst. 4, 285–306 (1985)

    Google Scholar 

  42. Pressley, A., Segal, G.: Loop Groups, Oxford Mathematical Monographs. Oxford (UK): Clarendon Press. viii, 318 p. (1988)

  43. Reed, M., Simon, B.: Functional Analysis (Methods of Modern Mathematical Physics Vol. I). Academic Press, Boca Raton (1980)

    Google Scholar 

  44. Reed, M., Simon, B.: Fourier Analysis, Self-Adjointness (Methods of Mathematical Physics Vol. II). Academic Press, Boca Raton (1975)

    Google Scholar 

  45. Restrepo, G.: Differentiable norms in Banach spaces. Bull. Am. Math. Soc. 70, 413–414 (1964)

    Google Scholar 

  46. Reyman, A.G., Semenov-Tian-Shansky, M.A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. Inventiones Math. 54, 81–100 (1979)

    Google Scholar 

  47. Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, Nonlinear partial differential equations in applied science, Proc. U.S.–Jap. Semin., Tokyo 1982, North-Holland Math. Stud. vol. 81, pp. 259–271 (1983)

  48. Schatten, R.: A theory of cross spaces. Annals of Mathematical Studies 26, Princeton N.J. (1950)

  49. Segal, G.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys. 80(3), 301–342 (1981)

    Google Scholar 

  50. Segal, G.: Loop groups and harmonic maps. Advances in homotopy theory, Proc. Conf. in Honour of I. M. James, Cortona/Italy 1988, Lond. Math. Soc. Lect. Note Ser 139, 153–164 (1989)

  51. Segal, G.: The geometry of the KdV equation. Int. J. Mod. Phys. A 6(16), 2859–2869 (1991)

    Google Scholar 

  52. Segal, G., Wilson, G.: Loop Groups and equations of KdV type. Pub. Math. I.H.E.S. 61, 5–65 (1985)

    Google Scholar 

  53. Semenov-Tian-Shansky, M.A.: What is a classical \(r\)-matrix? Funct. Anal. Appl. 17(4), 259–272 (1983)

    Google Scholar 

  54. Semenov-Tian-Shansky, M.A.: Dressing transformations and Poisson group actions. Publ. RIMS (Kyoto) 21, 1237–1260 (1985)

    Google Scholar 

  55. Semenov-Tian-Shansky, M.A.: Classical \(r\)-matrices, Lax equations, Poisson Lie Groups and Dressing Transformations, Field Theory, quantum gravity and string II (Meudon/Paris, 1985–1986). Lecture Notes in Physics, vol. 280. Springer, Berlin (1987)

  56. Semenov-Tian-Shansky, M. A.: Lectures on \(R\)-matrices, Poisson–Lie groups and integrable systems, In: Lectures on Integrable Systems, In Memory of J.-L. Verdier, Proc. of the CIMPA School on Integrable systems, Nice (France) 1991, O.Babelon, P. Cartier, Y. Kosmann-Schwarzbach, eds., World Scientific, 269–317 (1994)

  57. Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979)

    Google Scholar 

  58. Tumpach, A.B.: Banach Poisson–Lie group structure on \(\operatorname{U}({\cal{H}})\), (in preparation)

  59. Tumpach, A.B.: Variétés kählériennes et hyperkählériennes de dimension infinie, Thèse de doctorat, École polytechnique, Palaiseau, France (2005)

  60. Tumpach, A.B.: Hyperkähler structures and infinite-dimensional Grassmannians. J. Funct. Anal. 243, 158–206 (2007)

    Google Scholar 

  61. Weinstein, A.: The local structure of Poisson manifolds. J. Differ. Geom. 18, 523–557 (1983)

    Google Scholar 

  62. Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. (New Series) Am. Math. Soc. 16(1), 101–104 (1987)

    Google Scholar 

  63. Weinstein, A.: Some remarks on dressing transformations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35(1), 163–167 (1988)

    Google Scholar 

  64. Weinstein, A.: Poisson geometry. Differ. Geom. Appl. 9, 213–238 (1998)

    Google Scholar 

  65. Wurzbacher, T.: Fermionic Second Quantization and the Geometry of the Restricted Grassmannian, Infinite Dimensional Kähler Manifolds, DMV Seminar, Band 31, Birkhäuser, (2001)

  66. Zakharevich, I.: The second Gelfand-Dickey Bracket as a Bracket on a Poisson-Lie Grassmannian, Gelfand Seminar, Amer. Math. Soc, 1993, 179–208, Commun. Math. Phys. 159, 93–119 (1994)

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Acknowledgements

I would first and foremost like to thank T.S. Ratiu who brought the theory of Poisson–Lie groups to my attention when I was a post-graduate in Lausanne. It was only much later that I understood it to be the key to understanding [52] via the Poisson theory. I could not have written my paper without the help of D. Beltiţă who not only gave me the references indicating where the problem of triangulating an operator was studied, but who also gave me the electronic version of the documents when I was unable to go to any library. My paper was finally produced thanks to the support of the CNRS, of the University of Lille (France), in particular thanks to the CEMPI Labex (ANR-11-LABX-0007-01), as well as to the Pauli Institute in Vienna (Austria) offering very nice working conditions. The special assistance given by the CNRS for the promotion of women scientists was crucial for the inception of this paper which also benefited from the exchange in the early months of 2015 with other researchers during the Shape Analysis programme at the Erwin Schrodinger Institute in Vienna. The discussions with C. Vizman, K.-H. Neeb and F. Gay-Balmaz were particularly fruitful as were the WGMP lectures (exchanges with D. Beltiţă, T. Golinski, F. Pelletier, C. Roger and, last but not least, G. Larotonda). I am grateful to the anonymous referees for their pertinent comments which made me improve the quality of this paper. Finally I heartily thank Y. Kosmann-Schwarzbach and D. Bennequin for their valuable appreciations of a presentation of my paper.

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  1. Univ. Lille, CNRS, UMR 8524 Laboratoire Paul Painlevé, 59000, Lille, France

    A. B. Tumpach

  2. Wolfgang Pauli Institut, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    A. B. Tumpach

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Communicated by Y. Kawahigashi

Dedicated to T.S. Ratiu for his 70’s birthday and for the contagious enthusiasm he has in doing Mathematics.

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Tumpach, A.B. Banach Poisson–Lie Groups and Bruhat–Poisson Structure of the Restricted Grassmannian. Commun. Math. Phys. 373, 795–858 (2020). https://doi.org/10.1007/s00220-019-03674-3

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