Nondegenerate cohomology pairing for transitive Lie algebroids, characterization

Jan Kubarski; Alexandr Mishchenko

Open Mathematics (2004)

  • Volume: 2, Issue: 5, page 663-707
  • ISSN: 2391-5455

Abstract

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The Evens-Lu-Weinstein representation (Q A, D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q Aor, Dor) by tensoring by orientation flat line bundle, Q Aor=QA⊗or (M) and D or=D⊗∂Aor. It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q Aor, Dor) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂Aor) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.

How to cite

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Jan Kubarski, and Alexandr Mishchenko. "Nondegenerate cohomology pairing for transitive Lie algebroids, characterization." Open Mathematics 2.5 (2004): 663-707. <http://eudml.org/doc/268780>.

@article{JanKubarski2004,
abstract = {The Evens-Lu-Weinstein representation (Q A, D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q Aor, Dor) by tensoring by orientation flat line bundle, Q Aor=QA⊗or (M) and D or=D⊗∂Aor. It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q Aor, Dor) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂Aor) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.},
author = {Jan Kubarski, Alexandr Mishchenko},
journal = {Open Mathematics},
keywords = {58 H 99; 17 B 56; 18 G 40; 55 R 20; 55 T 05; 57 R 19; 57 R 22; 58 A 10},
language = {eng},
number = {5},
pages = {663-707},
title = {Nondegenerate cohomology pairing for transitive Lie algebroids, characterization},
url = {http://eudml.org/doc/268780},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Jan Kubarski
AU - Alexandr Mishchenko
TI - Nondegenerate cohomology pairing for transitive Lie algebroids, characterization
JO - Open Mathematics
PY - 2004
VL - 2
IS - 5
SP - 663
EP - 707
AB - The Evens-Lu-Weinstein representation (Q A, D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q Aor, Dor) by tensoring by orientation flat line bundle, Q Aor=QA⊗or (M) and D or=D⊗∂Aor. It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q Aor, Dor) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂Aor) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.
LA - eng
KW - 58 H 99; 17 B 56; 18 G 40; 55 R 20; 55 T 05; 57 R 19; 57 R 22; 58 A 10
UR - http://eudml.org/doc/268780
ER -

References

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  1. [1] R. Bott and L. Tu: Differential forms in algebraic topology, GTM 82, Springer-Verlag, 1982. Zbl0496.55001
  2. [2] S.S. Chern, F. Hirzebruch and J-P. Serre: “On the index of a fibered manifold”, Proc. AMS, Vol. 8, (1957), pp. 587–596. http://dx.doi.org/10.2307/2033523 Zbl0083.17801
  3. [3] M. Crainic: “Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes”, preprint, arXiv:math.DG/0008064, Commentarii Mathematici Helvetici, to appear. 
  4. [4] S. Evens, J-H. Lu and A. Weinstein: “Transverse measures, the modular class and a cohomology pairing for Lie algebroids”, Quart. J. Math. Oxford, Vol. 50, (1999), pp. 417–436. http://dx.doi.org/10.1093/qjmath/50.200.417 Zbl0968.58014
  5. [5] R.L. Fernandes: “Lie algebroids, holonomy and characteristic classes”, preprint DG/007132, Advances in Mathematics, Vol. 170, (2002), pp. 119–179. http://dx.doi.org/10.1006/aima.2001.2070 
  6. [6] F. Guedira and A. Lichnerowicz: “Géometrie des algébres de Lie locales de Kirillov”, J. Math. Pures Appl., Vol. 63, (1984), pp. 407–484. Zbl0562.53029
  7. [7] W. Greub, S. Halperin and R. Vanstone: Connections, Curvature and Cohomology, New York and London, Vol. I, 1971; Vol. II, 1973. Zbl0372.57001
  8. [8] S. Haller and T. Rybicki: “Reduction for locally conformal symplectic manifolds”, J. Geom. Phys., Vol. 37, (2001), pp. 262–271. http://dx.doi.org/10.1016/S0393-0440(00)00050-4 Zbl1005.53062
  9. [9] G. Hochschild and J.-P. Serre: “Cohomology of Lie algebras”, Ann. Math., Vol. 57, (1953), pp. 591–603. http://dx.doi.org/10.2307/1969740 Zbl0053.01402
  10. [10] J. Huebschmann: “Duality for Lie-Rinehart algebras and the modular class”, J. Reine Angew: Math, Vol. 510, (1999), pp. 103–159. Zbl1034.53083
  11. [11] V. Itskov, M. Karashev and Y. Vorobjev: “Infinitesimal Poisson Cohomology”, Amer. Math. Soc. Transl. (2), Vol. 187 (1998). 
  12. [12] J. Kubarski: “Invariant cohomology of regular Lie algebroids”, In: X. Masa, E. Macias-Virgos, J. Alvarez Lopez (Eds.): Proceedings of the VII International Colloquium on Differential Geometry ANALYSIS AND GEOMETRY IN FOLIATED MANIFOLD, Santiago de Compostela, Spain, July 1994, World Scientific, Singapure, 1995, pp. 137–151. Zbl0996.22005
  13. [13] J. Kubarski: “Fibre integral in regular Lie algebroids”, In: Proceedings of the Conference: New Developments in Differential Geometry, Budapest 1996, Budapest, Hungary, 27–30 July 1996, Kluwer Academic Publishers, 1999. 
  14. [14] J. Kubarski: “Poincaré duality for transitive unimodular invariantly oriented Lie algebroids”, Topology and Its Applications, Vol. 121(3), (2002), pp. 333–355. http://dx.doi.org/10.1016/S0166-8641(01)00102-X Zbl1048.58014
  15. [15] R. Kadobianski, J. Kubarski, V. Kushnirevitch and R. Wolak: “Transitive Lie algebroids of rank 1 and locally conformal symplectic structures”, Journal of Geometry and Physics, Vol. 46, (2003), pp. 151–158. http://dx.doi.org/10.1016/S0393-0440(02)00128-6 Zbl1034.53084
  16. [16] J. Kubarski and A.S. Mishchenko: “Lie algebroids: spectral sequences and signature”, Sbornik: Mathematics, Vol. 194(7), (2003), pp. 1079–1103. http://dx.doi.org/10.1070/SM2003v194n07ABEH000756 Zbl1074.58008
  17. [17] K. Mackenzie: Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc., Lecture Notes, Series 124, Cambridge University Press, 1987. Zbl0683.53029
  18. [18] M. Spivak: A Comprehensive Introduction to Differential Geometry, Vol. I, 2nd Ed., Publish or Perish Inc., Berkeley, 1979. 
  19. [19] A. Weinstein: “The modular automorphism group of a Poisson manifold”, J. Geom. Phys., Vol. 23, (1997), pp. 379–394. http://dx.doi.org/10.1016/S0393-0440(97)80011-3 Zbl0902.58013

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