Skip to main content
SpringerLink
Log in

On the Cohomology Ring of an Algebra

  • Chapter
Advances in Geometry

Part of the book series: Progress in Mathematics ((PM,volume 172))

Abstract

We define several versions of the cohomology ring of an associative algebra. These ring structures unify some well known operations from homological algebra and differential geometry. They have some formal resemblance with the quantum multiplication on Floer cohomology of free loop spaces. We discuss some examples, as well as applications to index theorems, characteristic classes and deformations.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.L. Brylinski, Some examples of Hochschild and cyclic homology Lecture Notes in Math. 1271(1987), 33–72

    Article  MathSciNet  Google Scholar 

  2. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowiscz, D. Sternheimer, Deformation theory and quantizationAnn. Phys. 111(1978), 61–151

    Article  MATH  Google Scholar 

  3. J.L Brylinski and E. Getzler, The homology of algebras of pseudo-differenial symbols and the noncommutative residueK-Theory 1(1987), 385–403

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Cartan and S. EilenbergHomological AlgebraPrinceton Uni-versity Press, 1956

    MATH  Google Scholar 

  5. A. Connes, Noncommutative differential geometry, Publ. Math. IRES, 1986

    Google Scholar 

  6. A. Connes, M. Flato, D. Sternheimer, Closed star products and cyclic homology, preprint, Paris, 1991

    Google Scholar 

  7. Y. Daletsky and B. Tsygan, Operations on Hochschild and cyclic complexes, preprint 1992

    Google Scholar 

  8. M. Deligne, Déformations d’algèbre des fonctions d’une variété symplectique: comparaison entre Fedosov et DeWilde, LecomteSelecta Math. 1(1995), 667–697

    Article  MathSciNet  MATH  Google Scholar 

  9. M. De Wilde and P.B.A. Lecomte, Existence of star-products and of formal deformations in Poisson Lie algebras of arbitrary Poisson manifoldsLett. Math. Phys. 7(1983), 487–496

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Fedosov, Index theoremsItogi Nauki i Tekhniki 65(1991), (in Russian)

    Google Scholar 

  11. B. Feigin and B. Tsygan, Additive K-theoryLecture Notes in Math.1289(1987), 66–220

    Google Scholar 

  12. K. Fukaya, Morse homotopyAcategories and Floer homolo-gies, MSRI preprint No. 020–94, 1993.

    Google Scholar 

  13. M. Gerstenhaber, The cohomology structure of an associative ring, Ann.of Math.78(1963), 267–288

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Gerstenhaber and A. Voronov, Homotopy G-algebras and moduli space operadInt. Math. Res. Notices(1995), 141–153

    Google Scholar 

  15. E. Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homologyIsrael Math. Conf. Proc. 7(1993), 65–78

    MathSciNet  Google Scholar 

  16. E. Getzler and J. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, preprint hep-th/9403055, 1994

    Google Scholar 

  17. E. Getzler and J. Jones, A, algebras and the cyclic bar complexIllinois J. of Math. 34(1990), 256–283

    MathSciNet  MATH  Google Scholar 

  18. E. Getzler, J. Jones and S. Petrack, Differential forms on loop spaces and the cyclic bar complexTopology 30(1991), 339–371

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Getzler and A. Szenes, On the Chern character of a theta-summable Fredholm moduleJ. Funct. Anal. 84(1989), 343–357

    Article  MathSciNet  MATH  Google Scholar 

  20. C.E. Hood and J. Jones, Some algebraic properties of cyclic ho-mology groupsK-theory 1(1987), 361–384

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Hochschild, B. Kostant and A. Rosenberg, Differential forms on regular affine algebrasTransactions AMS 102(1962), 383–408

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Jaffe, A. Lesniewski and K. Osterwalder, Quantum K-theoryI K-theory 2(1989), 675–682

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Kontsevich, Deformation quantization of Poisson manifolds, QA/9709040

    Google Scholar 

  24. J.L. Loday and D. Quillen, Cyclic homology and Lie alge-bra homology of matricesComment. Math. Helv. 59(1984), 565–591

    Article  MathSciNet  MATH  Google Scholar 

  25. Yu. I. Manin, Topics in non-commutative geometry, Princeton University Press, 1991

    Google Scholar 

  26. J.L. Loday, Cyclic Homology, Springer Verlag, 1993

    Google Scholar 

  27. D. MacDuff and D. Salomon, Quantum Comology and Jholomorphic Curves, AMS University Lecture Series, 1994

    Google Scholar 

  28. R. Nest and B. Tsygan, Algebraic index theorem for familiesAdvances in Math 113(1995), 151–205

    Article  MathSciNet  MATH  Google Scholar 

  29. R. Nest and B. Tsygan, Fukaya type categories for associative algebras, in:Deformation Theory and Symplectic GeometryMath. Phys. Studies, vol. 20, Kluwer Acad. Publ. (1997)

    Google Scholar 

  30. R. Nest and B. Tsygan, Homological properties of the category of algebras and characteristic classes, Preprint, Univ. of Heidelberg, 1995

    Google Scholar 

  31. G. Rinehart, Differential forms on commutative algebras, Trans. AMS 108(1963), 139–174

    Article  MathSciNet  Google Scholar 

  32. B.Tsygan, Homology of matrix Lie algebras over rings and Hochschild homologyUspekhi Mat. Nauk 38:2 (1983), 217–218

    MathSciNet  MATH  Google Scholar 

  33. M. Wodzicki, Cyclic homology of differential operatorsDuke Mathematical Journal 54(1987), 641–647

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Wodzicki, Cyclic homology of pseudodifferential operators and noncommutative Euler classC.R.A.S. 306(1988), 321–325

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Copenhagen, 5 Universitätsparken, Copenhagen, 2100, Denmark

    Ryzard Nest

  2. Department of Mathematics, Penn State University, University Park, PA, 16803, USA

    Boris Tsygan

Authors
  1. Ryzard Nest
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris Tsygan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Jean-Luc Brylinski , Ranee Brylinski , Victor Nistor , Boris Tsygan  & Ping Xu , , ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media New York

About this chapter

Cite this chapter

Nest, R., Tsygan, B. (1999). On the Cohomology Ring of an Algebra. In: Brylinski, JL., Brylinski, R., Nistor, V., Tsygan, B., Xu, P. (eds) Advances in Geometry. Progress in Mathematics, vol 172. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1770-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1770-1_14

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7274-8

  • Online ISBN: 978-1-4612-1770-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation