Mots-clés : Banach Lie–Poisson spaces
@article{SIGMA_2024_20_a103,
author = {Tomasz Goli\'nski and Alice Barbora Tumpach},
title = {Geometry of {Integrable} {Systems} {Related} to the {Restricted} {Grassmannian}},
journal = {Symmetry, integrability and geometry: methods and applications},
year = {2024},
volume = {20},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a103/}
}
TY - JOUR AU - Tomasz Goliński AU - Alice Barbora Tumpach TI - Geometry of Integrable Systems Related to the Restricted Grassmannian JO - Symmetry, integrability and geometry: methods and applications PY - 2024 VL - 20 UR - http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a103/ LA - en ID - SIGMA_2024_20_a103 ER -
Tomasz Goliński; Alice Barbora Tumpach. Geometry of Integrable Systems Related to the Restricted Grassmannian. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a103/
[1] Alvarado C.D., Chiumiento E., “Homogeneous spaces in Hartree–Fock–Bogoliubov theory”, J. Geom. Anal., 34 (2024), 334, 48 pp., arXiv: 2402.15606 | DOI | MR | Zbl
[2] Beltiţă D., Goliński T., Jakimowicz G., Pelletier F., “Banach–Lie groupoids and generalized inversion”, J. Funct. Anal., 276 (2019), 1528–1574, arXiv: 1802.09430 | DOI | MR | Zbl
[3] Beltiţă D., Ratiu T.S., “Symplectic leaves in real Banach Lie–Poisson spaces”, Geom. Funct. Anal., 15 (2005), 753–779, arXiv: math.SG/0403345 | DOI | MR | Zbl
[4] Beltiţă D., Ratiu T.S., Tumpach A.B., “The restricted Grassmannian, Banach Lie–Poisson spaces, and coadjoint orbits”, J. Funct. Anal., 247 (2007), 138–168, arXiv: math.DG/0606327 | DOI | MR | Zbl
[5] Bolsinov A.V., Borisov A.V., “Compatible Poisson brackets on Lie algebras”, Math. Notes, 72 (2002), 10–30 | DOI | MR | Zbl
[6] Bolsinov A.V., Konyaev A.Yu., Matveev V.S., “Applications of Nijenhuis geometry IV: Multicomponent KdV and Camassa–Holm equations”, Dyn. Partial Differ. Equ., 20 (2023), 73–98, arXiv: 2206.12942 | DOI | MR | Zbl
[7] Fomenko A.T., Trofimov V.V., Integrable systems on Lie algebras and symmetric spaces, Adv. Stud. Contemp. Math., 2, Gordon and Breach Science Publishers, New York, 1988 | MR
[8] Gay-Balmaz F., Ratiu T.S., Tumpach A.B., “The Restricted Siegel disc as coadjoint orbit”, Geometric Methods in Physics XL, Trends Math., Springer, Cham, 2024, 59–79, arXiv: 2405.13533 | DOI | MR
[9] Gay-Balmaz F., Ratiu T.S., Tumpach A.B., The hyperkaehler metric on the cotangent space of an infinite-dimensional symmetric Hilbert domain, in preparation
[10] Goliński T., Układy całkowalne na przestrzeniach Banacha Lie–Poissona zwia̧zanych z grassmannianem Sato, Ph.D. Thesis, Politechnika Warszawska, 2009 | Zbl
[11] Goliński T., Jakimowicz G., Sliżewska A., Banach Lie groupoid of partial isometries over restricted Grassmannian, arXiv: 2404.12847
[12] Goliński T., Odzijewicz A., “Some integrable systems on Banach Lie–Poisson space ${\rm i}\mathbb{R}\oplus \mathcal{U}^1_{\textrm{res}}$”, XXVIII Workshop on Geometric Methods in Physics, AIP Conf. Proc., 1191, American Institute of Physics, 2009, 91–97, arXiv: 2311.02449 | DOI | Zbl
[13] Goliński T., Odzijewicz A., “Hierarchy of Hamilton equations on Banach Lie–Poisson spaces related to restricted Grassmannian”, J. Funct. Anal., 258 (2010), 3266–3294, arXiv: 0908.2738 | DOI | MR | Zbl
[14] Goliński T., Tumpach A.B., “Integrable system on partial isometries: a finite-dimensional picture”, Geometric Methods in Physics XL, Trends Math., Springer, Cham, 2024, 49–57, arXiv: 2311.07412 | DOI | MR
[15] Hitchin N.J., Segal G.B., Ward R.S., Integrable systems, Oxf. Grad. Texts Math., 4, The Clarendon Press, Oxford University Press, New York, 1999 | MR
[16] Kalton N.J., “Trace-class operators and commutators”, J. Funct. Anal., 86 (1989), 41–74 | DOI | MR | Zbl
[17] Laurent-Gengoux C., Pichereau A., Vanhaecke P., Poisson structures, Grundlehren Math. Wiss., 347, Springer, Heidelberg, 2013 | DOI | MR
[18] Magri F., “A simple model of the integrable Hamiltonian equation”, J. Math. Phys., 19 (1978), 1156–1162 | DOI | MR | Zbl
[19] Marsden J.E., Ratiu T.S., Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, Texts Appl. Math., 17, 2nd ed., Springer, New York, 1999 | DOI | MR | Zbl
[20] Mickelsson J., Current algebras and groups, Plenum Monogr. Nonlinear Phys., Plenum Press, New York, 1989 | DOI | MR | Zbl
[21] Odzijewicz A., Goliński T., “Hierarchy of integrable Hamiltonians describing the nonlinear $n$-wave interaction”, J. Phys. A, 45 (2012), 045204, 14 pp., arXiv: 1106.3217 | DOI | MR | Zbl
[22] Odzijewicz A., Ratiu T.S., “Banach Lie–Poisson spaces and reduction”, Comm. Math. Phys., 243 (2003), 1–54, arXiv: math.SG/0210207 | DOI | MR | Zbl
[23] Odzijewicz A., Ratiu T.S., “Extensions of Banach Lie–Poisson spaces”, J. Funct. Anal., 217 (2004), 103–125, arXiv: math.SG/0310312 | DOI | MR | Zbl
[24] Odzijewicz A., Ratiu T.S., “Induced and coinduced Banach Lie–Poisson spaces and integrability”, J. Funct. Anal., 255 (2008), 1225–1272 | DOI | MR | Zbl
[25] Odzijewicz A., Sliżewska A., “Banach–Lie groupoids associated to $W^*$-algebras”, J. Symplectic Geom., 14 (2016), 687–736 | DOI | MR | Zbl
[26] Pressley A., Segal G., Loop groups, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1986 | MR
[27] Sato M., Sato Y., “Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold”, Nonlinear Partial Differential Equations in Applied Science, North-Holland Math. Stud., 81, North-Holland, Amsterdam, 1983, 259–271 | DOI | MR | Zbl
[28] Schwinger J., “Field theory commutators”, Phys. Rev. Lett., 3 (1959), 296–297 | DOI
[29] Segal G., Wilson G., “Loop groups and equations of KdV type”, Inst. Hautes Études Sci. Publ. Math., 61 (1985), 5–65 | DOI | MR | Zbl
[30] Spera M., Valli G., “Plücker embedding of the Hilbert space Grassmannian and the CAR algebra”, Russian J. Math. Phys., 2 (1994), 383–392 | MR | Zbl
[31] Tumpach A.B., “Hyperkähler structures and infinite-dimensional Grassmannians”, J. Funct. Anal., 243 (2007), 158–206, arXiv: math-ph/0511056 | DOI | MR | Zbl
[32] Tumpach A.B., “Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits”, Ann. Inst. Fourier (Grenoble), 59 (2009), 167–197, arXiv: math-ph/0605032 | DOI | MR | Zbl
[33] Tumpach A.B., “On the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits”, Forum Math., 21 (2009), 375–393, arXiv: math-ph/0703042 | DOI | MR | Zbl
[34] Tumpach A.B., “Banach Poisson–Lie groups and Bruhat–Poisson structure of the restricted Grassmannian”, Comm. Math. Phys., 373 (2020), 795–858, arXiv: 1805.03292 | DOI | MR | Zbl
[35] Wurzbacher T., “Fermionic second quantization and the geometry of the restricted Grassmannian”, Infinite Dimensional Kähler Manifolds (Oberwolfach, 1995), DMV Sem., 31, Birkhäuser, Basel, 2001, 287–375 | DOI | MR | Zbl