On the index of nonlocal elliptic operators for compact Lie groups

• Autorzy: Anton Savin
• Języki publikacji: EN

Abstrakty

EN

We consider a class of nonlocal operators associated with a compact Lie group G acting on a smooth manifold. A notion of symbol of such operators is introduced and an index formula for elliptic elements is obtained. The symbol in this situation is an element of a noncommutative algebra (crossed product by G) and to obtain an index formula, we define the Chern character for this algebra in the framework of noncommutative geometry.

Słowa kluczowe

EN

Nonlocal operator; Basic cohomology; Crossed product; Transverse ellipticity; Chern character; Index formula

Kategorie tematyczne

• 19K56: Index theory
• 46L87: Noncommutative differential geometry
• 58J20: Index theory and related fixed point theorems

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 4
• Strony: 833-850

Daty

wydano

2011-08-01

online

2011-05-26

Twórcy

autor

Anton Savin

Bibliografia

• [1] Antonevich A.B., Elliptic pseudodifferential operators with a finite group of shifts, Izv. Akad. Nauk SSSR Ser. Mat., 1973, 37(3), 663–675 (in Russian)
• [2] Atiyah M.F., Elliptic Operators and Compact Groups, Lecture Notes in Math., 401, Springer, Berlin-New York, 1974
• [3] Atiyah M.F., Segal G.B., The index of elliptic operators II, Ann. of Math., 1968, 87(3), 531–545 http://dx.doi.org/10.2307/1970716
• [4] Baum P., Connes A., Chern character for discrete groups, In: A Fête of Topology, Academic Press, Boston, 1988, 163–232
• [5] Berline N., Getzler E., Vergne M., Heat Kernels and Dirac Operators, Grundlehren Math. Wiss., 298, Springer, Berlin, 1992
• [6] Berline N., Vergne M., The Chern character of a transversally elliptic symbol and the equivariant index, Invent. Math., 1996, 124(1–3), 11–49 http://dx.doi.org/10.1007/s002220050045
• [7] Block J., Getzler E., Equivariant cyclic homology and equivariant differential forms, Ann. Sci. École Normale Sup., 1994, 27(4), 493–527
• [8] Bredon G.E., Introduction to Compact Transformation Groups, Pure Appl. Math., 46, Academic Press, New York-London, 1972
• [9] Brylinski J.-L., Nistor V., Cyclic cohomology of étale groupoids, K-Theory, 1994, 8(4), 341–365 http://dx.doi.org/10.1007/BF00961407
• [10] Connes A., Cyclic cohomology and the transverse fundamental class of a foliation, In: Geometric Methods in Operator Algebras, Kyoto, 1983, Pitman Res. Notes Math. Ser., 123, Longman, Harlow, 1986, 52–144
• [11] Connes A., Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math., 1985, 62, 257–360 http://dx.doi.org/10.1007/BF02698807
• [12] Dave S., An equivariant noncommutative residue, preprint available at http://arxiv.org/abs/math/0610371
• [13] Dave S., Equivariant homology for pseudodifferential operators, preprint available at http://arxiv.org/abs/1005.2282
• [14] Karoubi M., Homologie Cyclique et K-théorie, Astérisque, 149, Soc. Math. France Inst. Henri Poincaré, Paris, 1987
• [15] Kawasaki T., The index of elliptic operators over V-manifolds, Nagoya Math. J., 1981, 84, 135–157
• [16] Kordyukov Yu.A., Index theory and non-commutative geometry on foliated manifolds, Russian Math. Surveys, 2009, 64(2), 273–391 http://dx.doi.org/10.1070/RM2009v064n02ABEH004616
• [17] Koszul J.L., Sur certains groupes de transformations de Lie, In: Géométrie Différentielle, Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre National de la Recherche Scientifique, Paris, 1953, 137–141
• [18] Nazaikinskii V.E., Savin A.Yu., Sternin B.Yu., Elliptic Theory and Noncommutative Geometry, Oper. Theory Adv. Appl., 183, Birkhäuser, Basel, 2008
• [19] Nistor V., Higher index theorems and the boundary map in cyclic cohomology, Doc. Math., 1997, 2, 263–295
• [20] Pedersen G.K., C*-Algebras and Their Automorphism Groups, London Math. Soc. Monogr., 14, Academic Press, London-New York, 1979
• [21] Quillen D., Superconnections and the Chern character, Topology, 1985, 24(1), 89–95 http://dx.doi.org/10.1016/0040-9383(85)90047-3
• [22] Sternin B.Yu., On a class of nonlocal elliptic operators for compact Lie groups. Uniformization and finiteness theorem, Cent. Eur. J. Math., 2011, 9(4)
• [23] Tsygan B.L., The homology of matrix Lie algebras over rings and the Hochschild homology, Russian Math. Surveys, 1983, 38(2), 198–199 http://dx.doi.org/10.1070/RM1983v038n02ABEH003481
• [24] Vergne M., Equivariant index formulas for orbifolds, Duke Math. J., 1996, 82(3), 637–652 http://dx.doi.org/10.1215/S0012-7094-96-08226-5
• [25] Verona A., A de Rham type theorem for orbit spaces, Proc. Amer. Math. Soc., 1988, 104(1), 300–302 http://dx.doi.org/10.1090/S0002-9939-1988-0958087-8

Identyfikatory

DOI

10.2478/s11533-011-0028-9