Skip to main content
SpringerLink
Log in

From Atiyah Classes to Homotopy Leibniz Algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes T X [−1] into a Lie algebra object in D + (X), the bounded below derived category of coherent sheaves on X. Furthermore, Kapranov proved that, for a Kähler manifold X, the Dolbeault resolution \({\Omega^{\bullet-1}(T_X^{1, 0})}\) of T X [−1] is an L ∞ algebra. In this paper, we prove that Kapranov’s theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair (L, A), i.e. a Lie algebroid L together with a Lie subalgebroid A, we define the Atiyah class α E of an A-module E as the obstruction to the existence of an A-compatible L-connection on E. We prove that the Atiyah classes α L/A and α E respectively make L/A[−1] and E[−1] into a Lie algebra and a Lie algebra module in the bounded below derived category \({D^+(\mathcal{A})}\) , where \({\mathcal{A}}\) is the abelian category of left \({\mathcal{U}(A)}\) -modules and \({\mathcal{U}(A)}\) is the universal enveloping algebra of A. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of L/A and E, and inducing the aforesaid Lie structures in \({D^+(\mathcal{A})}\) .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ammar M., Poncin N.: Coalgebraic approach to the Loday infinity category, stem differential for 2n-ary graded and homotopy algebras. Ann. Inst. Fourier (Grenoble) 60(1), 355–387 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arinkin, D., Căldăraru, A.: When is the self-intersection of a subvariety a fibration? Adv. Math. 231(2), 815–842 (2012)

  3. Atiyah M.F.: Complex analytic connections in fibre bundles. Trans. Am. Math. Soc. 85, 181–207 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bar-Natan D., Garoufalidis S., Rozansky L., Thurston D.P.: Wheels, wheeling, and the Kontsevich integral of the unknot. Israel J. Math. 119, 217–237 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bar-Natan, D., Le, T.T.Q., Thurston, D.P.: Two applications of elementary knot theory to Lie algebras and Vassiliev invariants. Geom. Topol. 7, 1–31 (2003)

  6. Bordemann, M.: (Bi)modules,morphisms, and reduction of star-products: the symplectic case, foliations, and obstructions. Trav. Math. 16, 9–40 (2005)

  7. Bordemann, M.: Atiyah classes and equivariant connections on homogeneous spaces. Trav. Math. 20, 29–82 (2012)

  8. Bott, R.: Lectures on characteristic classes and foliations, Lectures on algebraic and differential topology (Second Latin American School in Math., Mexico City, 1971), Springer, Berlin, Notes by Lawrence Conlon, with two appendices by J. Stasheff, pp. 1–94. Lecture Notes in Math., Vol. 279, (1972)

  9. Calaque D.: A PBW theorem for inclusions of (sheaves of) Lie algebroids. Rend. Semin. Mat. Univ. Padova 131, 23–47 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Calaque D., Căldăraru A., Tu J.: PBW for an inclusion of Lie algebras. J. Algebra 378, 64–79 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Calaque, D., Rossi, C.A.: Lectures on Duflo isomorphisms in Lie algebra and complex geometry, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich (2011)

  12. Calaque D., Van den Bergh M.: Hochschild cohomology and Atiyah classes. Adv. Math. 224(5), 1839–1889 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen Z., Stiénon M., Xu P.: A Hopf algebra associated with a Lie pair. C. R. Math. Acad. Sci. Paris 352(11), 929–933 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  14. Costello, K.: A geometric construction of the Witten genus, I. In: Proceedings of the international congress of mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 942–959

  15. Costello, K.: A geometric construction of the Witten genus, II. arXiv:1112.0816 (2011)

  16. Căldăraru A., Willerton S.: The Mukai pairing. I. A categorical approach. N Y J. Math. 16, 61–98 (2010)

    MATH  MathSciNet  Google Scholar 

  17. Dolgushev, V., Tamarkin, D., Tsygan, B.: Formality of the homotopy calculus algebra of hochschild (co)chains. arXiv:0807.5117 (2008)

  18. Dolgushev V., Tamarkin D., Tsygan B.: Formality theorems for Hochschild complexes and their applications. Lett. Math. Phys. 90(1–3), 103–136 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Grinberg, D.: Poincaré–Birkhoff–Witt type results for inclusions of Lie algebras. Master’s thesis, Massachusetts Institute of Technology. Boston, MA (2011)

  20. Higgins P.J., Mackenzie K.C.H.: Algebraic constructions in the category of Lie algebroids. J. Algebra 129(1), 194–230 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hübschmann J.: Poisson cohomology and quantization. J. Reine Angew. Math. 408, 57–113 (1990)

    MATH  MathSciNet  Google Scholar 

  22. Joyal A., Street R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kapranov M.: Rozansky–Witten invariants via Atiyah classes. Compos. Math. 115(1), 71–113 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48(1), 35–72 (1999), Moshé Flato (1937–1998)

  25. Kontsevich M.: Rozansky–Witten invariants via formal geometry. Compos. Math. 115(1), 115–127 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Lada, T.: L ∞ algebra representations. Appl. Categ. Struct. 12(1), 29–34 (2004)

  28. Lada T., Markl M.: Strongly homotopy Lie algebras. Commun. Algebra 23(6), 2147–2161 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lada T., Stasheff J.: Introduction to SH Lie algebras for physicists. Int. J. Theor. Phys. 32(7), 1087–1103 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  30. Laurent-Gengoux, C., Stiénon, M., Xu, P.: Holomorphic poisson manifolds and holomorphic Lie algebroids. Int. Math. Res. Not. IMRN (2008), Art. ID rnn 088, 46

  31. Laurent-Gengoux C., Stiénon M., Xu P.: Exponential map and L ∞ algebra associated to a Lie pair. C. R. Math. Acad. Sci. Paris 350(17–18), 817–821 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. Laurent-Gengoux, C., Stiénon, M., Xu, P.: Poincaré–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds. arXiv:1408.2903 (2014)

  33. Laurent-Gengoux, C., Voglaire, Y.: Invariant connections and PBW theorem for Lie groupoid pairs. arXiv:1507.01051 (2015)

  34. Loday J.-L.: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math. (2) 39(3–4), 269–293 (1993)

    MATH  MathSciNet  Google Scholar 

  35. Lu J.-H.: Poisson homogeneous spaces and Lie algebroids associated to Poisson actions. Duke Math. J. 86(2), 261–304 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  36. Mackenzie K.C.H., Mokri T.: Locally vacant double Lie groupoids and the integration of matched pairs of Lie algebroids. Geom. Dedicata 77(3), 317–330 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mackenzie K.C.H., Xu P.: Classical lifting processes and multiplicative vector fields. Q. J. Math. Oxf. Ser. (2) 49(193), 59–85 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  38. Markarian N.: The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem. J. Lond. Math. Soc. (2) 79(1), 129–143 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  39. Mehta R.A., Stiénon M., Xu P.: The Atiyah class of a dg-vector bundle. C. R. Math. Acad. Sci. Paris 353(4), 357–362 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  40. Mokri T.: Matched pairs of Lie algebroids. Glasgow Math. J. 39(2), 167–181 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  41. Molino P.: Classe d’Atiyah d’un feuilletage et connexions transverses projetables. C. R. Acad. Sci. Paris Sér. A-B 272, A779–A781 (1971)

    MathSciNet  MATH  Google Scholar 

  42. Molino P.: Propriétés cohomologiques et propriétés topologiques des feuilletages à à connexion transverse projetable. Topology 12, 317–325 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  43. Ramadoss A.C.: The big Chern classes and the Chern character. Int. J. Math. 19(6), 699–746 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  44. Rinehart G.S.: Differential forms on general commutative algebras. Trans. Am. Math. Soc. 108, 195–222 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  45. Roberts J., Willerton S.: On the Rozansky–Witten weight systems. Algebr. Geom. Topol. 10(3), 1455–1519 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  46. Rozansky L., Witten E.: Hyper–Kähler geometry and invariants of three-manifolds. Selecta Math. (N.S.) 3(3), 401–458 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  47. Shoikhet, B.: On the Duflo formula for L ∞-algebras and Q-manifolds. arXiv:math/9812009 (1998)

  48. Stasheff J.: Constrained Poisson algebras and strong homotopy representations. Bull. Am. Math. Soc. (N.S.) 19(1), 287–290 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  49. Uchino K.: Derived brackets and sh Leibniz algebras. J. Pure Appl. Algebra 215(5), 1102–1111 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  50. Vaisman I.: Remarks about differential operators on foliate manifolds. An. Şti. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. (N.S.) 20(2), 327–350 (1974)

    MATH  MathSciNet  Google Scholar 

  51. Vitagliano, L.: On the strong homotopy Lie–Rinehart algebra of a foliation. Commun. Contemp. Math. 16(6), 1450007, 49 (2014)

  52. Vitagliano, L.: On the strong homotopy associative algebra of a foliation. Commun. Contemp. Math. 17(2), 1450026, 34 (2015)

  53. Vitagliano L.: Representations of homotopy Lie–Rinehart algebras. Math. Proc. Camb. Philos. Soc. 158(1), 155–191 (2015)

    Article  MathSciNet  Google Scholar 

  54. Voglaire Y., Xu P.: Rozansky–Witten-type invariants from symplectic Lie pairs. Commun. Math. Phys. 336(1), 217–241 (2015)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  55. Xu P.: Quantum groupoids. Commun. Math. Phys. 216(3), 539–581 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  56. Yu, S.: Dolbeault dga of formal neighborhoods and L ∞ algebroids, PhD thesis, Penn State University. State College, PA (2013)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tsinghua University, Beijing, China

    Zhuo Chen

  2. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Mathieu Stiénon & Ping Xu

Authors
  1. Zhuo Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mathieu Stiénon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ping Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ping Xu.

Additional information

Communicated by N. Reshetikhin

Research partially supported by NSFC Grant 11471179, the Beijing high education young elite teacher project, NSA Grant H98230-12-1-0234, and NSF Grants DMS0605725, DMS0801129, DMS1101827.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Z., Stiénon, M. & Xu, P. From Atiyah Classes to Homotopy Leibniz Algebras. Commun. Math. Phys. 341, 309–349 (2016). https://doi.org/10.1007/s00220-015-2494-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2494-6

Keywords

Search

Navigation