Skip to main content
SpringerLink
Log in

Combinatorics and Formal Geometry of the Maurer–Cartan Equation

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

Abstract

We give a general treatment of the Maurer–Cartan equation in homotopy algebras and describe the operads and formal differential geometric objects governing the corresponding algebraic structures. We show that the notion of Maurer–Cartan twisting is encoded in certain automorphisms of these universal objects.

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. Block, J.: Duality and equivalence of module categories in noncommutative geometry. A celebration of the mathematical legacy of Raoul Bott, pp. 311–339, CRM Proc. Lecture Notes, vol. 50. American Mathematical Society, Providence (2010)

  2. Chuang J., Lazarev A.: L-infinity maps and twistings. Homol. Homotopy Appl. 13(2), 175–195 (2011) arXiv:0912.1215

    Article  MathSciNet  MATH  Google Scholar 

  3. Dolgushev, V., Rogers, C.: Notes on Algebraic Operads, Graph Complexes, and Willwacher’s Construction. arXiv:math/0404003v4

  4. Getzler E.: Lie theory for nilpotent L-infinity algebras. Ann. Math. 170(1), 271–301 (2009) arXiv:math/0404003v4

    Article  MathSciNet  MATH  Google Scholar 

  5. Getzler, E., Kapranov, M.M.: Cyclic operads and cyclic homology. Geometry, topology, & physics, pp. 167–201. In: Conf. Proc. Lecture Notes Geom. Topology, IV. Int. Press, Cambridge (1995)

  6. Getzler E., Kapranov M.M.: Modular operads. Compos. Math. 110(1), 65–126 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ginzburg V., Kapranov M.: Koszul duality for operads. Duke Math. J. 76(1), 203–272 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hamilton, A., Lazarev, A.: Homotopy algebras and noncommutative geometry. arXiv: math/0410621

  9. Kontsevich, M.: Formal noncommutative symplectic geometry. In: The Gelfand Mathematical Seminars, 1990–1992, pp. 173–187. Birkhäuser, Boston (1993)

  10. Lefschetz, S.: Algebraic Topology. AMS, Colloquium Pbns. Series, vol. 27 (1942)

  11. Merkulov S.: An L ∞-algebra of an unobstructed deformation functor. Int. Math. Res. Notices 3, 147–164 (2000)

    Article  MathSciNet  Google Scholar 

  12. Reutenauer, C.: Free Lie algebras. In: London Mathematical Society Monographs. New Series, vol. 7. Oxford Science Publications. The Clarendon Press, New York (1993)

  13. Stasheff, J.: Deformation theory and the Batalin-Vilkovisky master equation. Deformation theory and symplectic geometry (Ascona, 1996), pp. 271–284. In: Math. Phys. Stud., vol. 20. Kluwer, Dordrecht (1997)

  14. Willwacher, T.: M. Kontsevich’s graph complex and the Grothendieck-Teichmueller Lie algebra. arxiv:1009.1654.

  15. Willwacher, T.: A Note on Br-infinity and KS-infinity Formality. arXiv:1109.3520

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematical Science, City University London, London, EC1V 0HB, UK

    Joseph Chuang

  2. Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

    Andrey Lazarev

Authors
  1. Joseph Chuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrey Lazarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joseph Chuang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chuang, J., Lazarev, A. Combinatorics and Formal Geometry of the Maurer–Cartan Equation. Lett Math Phys 103, 79–112 (2013). https://doi.org/10.1007/s11005-012-0586-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-012-0586-1

Mathematics Subject Classification (1991)

Keywords

Search

Navigation