Using a nonlinear version of the tautological bundle over Graßmannians, we construct a transgression map for differential characters from to the nonlinear Graßmannian of submanifolds of of a fixed type . In particular, we obtain prequantum circle bundles of the nonlinear Graßmannian endowed with the Marsden–Weinstein symplectic form. The associated Kostant–Souriau prequantum extension yields central Lie group extensions of a group of volume-preserving diffeomorphisms integrating Lichnerowicz cocycles.
En utilisant une version non-linéaire du fibré tautologique sur les Graßmanniennes, nous construisons une application de transgression pour les caractères différentiels de à la Graßmannienne non-linéaire des sous-variétés de d’un type fixé . En particulier, nous obtenons des fibrés en cercles préquantiques au dessus de la Graßmannienne non-linéaire doté de la forme symplectique de Marsden–Weinstein. L’extension préquantique de Kostant–Souriau associée donne des extensions centrales de groupes de Lie d’un groupe de difféomorphismes préservant le volume et intégrant les cocycles de Lichnerowicz.
Revised:
Accepted:
Published online:
Keywords: Volume-preserving diffeomorphism, Nonlinear Graßmannian, Tautological bundle, Differential character
Mot clés : Difféomorphisme préservant le volume, Grassmannienne non-linéaire, Fibré tautologique, Caractère différentiel
Diez, Tobias 1; Janssens, Bas 1; Neeb, Karl-Hermann 2; Vizman, Cornelia 3
@article{AIF_2024__74_6_2483_0, author = {Diez, Tobias and Janssens, Bas and Neeb, Karl-Hermann and Vizman, Cornelia}, title = {Induced differential characters on nonlinear {Gra{\ss}mannians}}, journal = {Annales de l'Institut Fourier}, pages = {2483--2503}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {74}, number = {6}, year = {2024}, doi = {10.5802/aif.3661}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3661/} }
TY - JOUR AU - Diez, Tobias AU - Janssens, Bas AU - Neeb, Karl-Hermann AU - Vizman, Cornelia TI - Induced differential characters on nonlinear Graßmannians JO - Annales de l'Institut Fourier PY - 2024 SP - 2483 EP - 2503 VL - 74 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3661/ DO - 10.5802/aif.3661 LA - en ID - AIF_2024__74_6_2483_0 ER -
%0 Journal Article %A Diez, Tobias %A Janssens, Bas %A Neeb, Karl-Hermann %A Vizman, Cornelia %T Induced differential characters on nonlinear Graßmannians %J Annales de l'Institut Fourier %D 2024 %P 2483-2503 %V 74 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3661/ %R 10.5802/aif.3661 %G en %F AIF_2024__74_6_2483_0
Diez, Tobias; Janssens, Bas; Neeb, Karl-Hermann; Vizman, Cornelia. Induced differential characters on nonlinear Graßmannians. Annales de l'Institut Fourier, Volume 74 (2024) no. 6, pp. 2483-2503. doi : 10.5802/aif.3661. https://aif.centre-mersenne.org/articles/10.5802/aif.3661/
[1] Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., Volume 53 (1978), pp. 174-227 | DOI | Zbl
[2] Differential Characters, Lecture Notes in Mathematics, 2112, Springer, 2014
[3] Loop Spaces, Characteristic Classes and Geometric Quantization, Modern Birkhäuser Classics, Birkhäuser, 2007
[4] On the group of automorphisms of a symplectic manifold, Problems in Analysis: A Symposium in Honor of Salomon Bochner (Princeton Mathematical Series), Volume 31, Princeton University Press, 1970
[5] Differential characters and geometric invariants, Geometry and Topology (College Park, Md., 1983/84) (Lecture Notes in Mathematics), Volume 1167, Springer, 1983, pp. 50-80 | Zbl
[6] Central extensions of Lie groups preserving a differential form, Int. Math. Res. Not., Volume 2021 (2021) no. 5, pp. 3794-3821 | DOI | Zbl
[7] Formally integrable complex structures on higher dimensional knot spaces, J. Symplectic Geom., Volume 19 (2021) no. 3, pp. 507-529 | DOI | Zbl
[8] Connections, curvature, and cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles, Pure and Applied Mathematics, 47, Academic Press Inc., 1972
[9] Nonlinear Graßmannians as coadjoint orbits, Math. Ann., Volume 329 (2004) no. 4, pp. 771-785 | DOI | Zbl
[10] Nonlinear flag manifolds as coadjoint orbits, Ann. Global Anal. Geom., Volume 58 (2020) no. 4, pp. 385-413 | DOI | Zbl
[11] Representations of infinite-dimensional groups, Translations of Mathematical Monographs, 152, American Mathematical Society, 1996 | DOI
[12] Representations of the Virasoro algebra by the orbit method, J. Geom. Phys., Volume 5 (1988) no. 3, pp. 351-363 | DOI | Zbl
[13] Quantization and unitary representations. I: Prequantization, Lectures in modern analysis and applications. III (Lecture Notes in Mathematics), Volume 170, Springer, 1970, pp. 87-208 | Zbl
[14] The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, 1997 | DOI
[15] An obstruction to removing intersection points in immersed surfaces, Topology, Volume 36 (1997) no. 4, pp. 931-962 | DOI | Zbl
[16] A Kähler structure on the space of string worldsheets, Class. Quant. Grav., Volume 10 (1993) no. 9, p. L141-L148 | DOI | Zbl
[17] Loop spaces as complex manifolds, J. Differ. Geom., Volume 38 (1993) no. 3, pp. 519-543 | Zbl
[18] Algèbre de Lie des automorphismes infinitésimaux d’une structure unimodulaire, Ann. Inst. Fourier, Volume 24 (1974) no. 3, pp. 219-266 | DOI | Numdam | Zbl
[19] Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D, Volume 7 (1983), pp. 305-323 | DOI | Zbl
[20] Infinite-dimensional groups and their representations, Lie theory. Lie algebras and representations (Progress in Mathematics), Volume 228, Birkhäuser, 2004, pp. 213-328 | Zbl
[21] Towards a Lie theory of locally convex groups, Jpn. J. Math., Volume 1 (2006) no. 2, pp. 291-468 | DOI | Zbl
[22] Flux homomorphisms and principal bundles over infinite dimensional manifolds, Monatsh. Math., Volume 139 (2003) no. 4, pp. 309-333 | DOI | Zbl
[23] Loop Groups, Oxford University Press, 1986
[24] Extensions centrales d’algèbres et de groupes de Lie de dimension infinie,algèbre de Virasoro et généralisations, Rep. Math. Phys., Volume 35 (1995) no. 2-3, pp. 225-266 | DOI | Zbl
[25] Diffeomorphisms of the 2-sphere, Proc. Am. Math. Soc., Volume 10 (1959), pp. 621-626 | DOI | Zbl
[26] Structure des systèmes dynamiques. Maitrises de mathématique, Dunod, 1970
[27] A formally Kähler structure on a knot space of a -manifold, Sel. Math., New Ser., Volume 18 (2012) no. 3, pp. 539-555 | DOI | Zbl
[28] Lichnerowicz cocycles and central Lie group extensions, An. Univ. Vest Timiș., Ser. Mat.-Inform., Volume 48 (2010) no. 1-2, pp. 285-297 | Zbl
[29] Induced differential forms on manifolds of functions, Arch. Math., Brno, Volume 47 (2011) no. 3, pp. 201-215 | Zbl
Cited by Sources: