Abstract:
In this article we introduce a new language to describe many problems of differential geometry: for example, problems connected with the theory of pseudogroups, Lie equations, foliations, characteristic classes, etc. This is the language of infinite-dimensional Lie algebras and their homogeneous spaces. It is closely connected with the general idea of formal differential geometry set forth by I. M. Gel'fand in his lecture at the International Congress of Mathematicians in Nice. In addition to a detailed account of the theory of homogeneous spaces of infinite-dimensional Lie algebras, this article contains applications of this theory to the characteristic classes of foliations. It also includes results on these questions from earlier papers by I. M. Gel'fand, D. A. Kazhdan, D. B. Fuks, and the authors.
Citation:
J. H. Bernstein, B. I. Rozenfel'd, “Homogeneous spaces of infinite-dimensional Lie algebras and characharacteristic classes of foliations”, Russian Math. Surveys, 28:4 (1973), 107–142
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A. Zuevsky, “Cosimplicial cohomology of restricted meromorphic functions on foliated manifolds”, Indagationes Mathematicae, 2024
Yaroslav V. Bazaikin, Anton S. Galaev, “LOSIK CLASSES FOR CODIMENSION-ONE FOLIATIONS”, J. Inst. Math. Jussieu, 21:4 (2022), 1391
Jean-Pierre Magnot, Enrique G. Reyes, “Well-Posedness of the Kadomtsev–Petviashvili Hierarchy, Mulase Factorization, and Frölicher Lie Groups”, Ann. Henri Poincaré, 21:6 (2020), 1893
Batu Güneysu, Markus J. Pflaum, “The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs”, SIGMA, 13 (2017), 003, 44 pp.
A. A. Bytsenko, M. Chaichian, R. J. Szabo, A. Tureanu, “Quantum black holes, elliptic genera and spectral partition functions”, Int. J. Geom. Methods Mod. Phys, 2014, 1450048
Gwilliam O., Grady R., “One-Dimensional Chern–Simons Theory and the (a)Over-Cap Genus”, Algebr. Geom. Topol., 14:4 (2014), 2299–2377
R. Hernandez Heredero, E. G. Reyes, “Geometric Integrability of the Camassa-Holm Equation. II”, International Mathematics Research Notices, 2011
M. V. Losik, “On the continuous cohomology of diffeomorphism groups”, Mosc. Math. J., 10:2 (2010), 377–397
Yu. A. Kordyukov, “Index theory and non-commutative geometry on foliated manifolds”, Russian Math. Surveys, 64:2 (2009), 273–391
Merkulov S.A., “Operad of Formal Homogeneous Spaces and Bernoulli Numbers”, Algebr. Number Theory, 2:4 (2008), 407–433
N. I. Zhukova, “Minimal Sets of Cartan Foliations”, Proc. Steklov Inst. Math., 256 (2007), 105–135
G. N. Bushueva, V. V. Shurygin, “On the higher order geometry of Weil bundles over smooth manifolds and over parameter-dependent manifolds”, Lobachevskii J. Math., 18 (2005), 53–105
V. A. Yumaguzhin, “Finite-type integrable geometric structures”, J. Math. Sci., 136:6 (2006), 4401–4410
Alberto S. Cattaneo, Giovanni Felder, Lorenzo Tomassini, “From local to global deformation quantization of Poisson manifolds”, Duke Math. J., 115:2 (2002)
B. M. Dubrov, B. P. Komrakov, “The problem of constructive equivalence in differential geometry”, Sb. Math., 191:5 (2000), 655–681
“Ot redaktsii”, Matem. sb., 191:7 (2000), 160–160
M. Fliess, J. Lévine, P. Martin, P. Rouchon, “On a new differential geometric setting in nonlinear control”, Journal of Mathematical Sciences (New York), 83:4 (1997), 524–530
Alexander Reznikov, “Rationality of secondary classes”, J. Differential Geom., 43:3 (1996)
M. V. Losik, “A generalization of manifold and its characteristic classes”, Funct. Anal. Appl., 24:1 (1990), 26–32
M. V. Losik, “Characteristic classes of structures on manifolds”, Funct. Anal. Appl., 21:3 (1987), 206–216
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