Skip to main content
SpringerLink
Log in

Differential graded Lie algebras, quasi-hopf algebras and higher homotopy algebras

  • I. Quantum Groups, Deformation Theory And Representation Theory
  • Conference paper
  • First Online:
Quantum Groups

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1510))

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 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.95
Price excludes VAT (USA)
  • Compact, lightweight 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.F.Adams, On the cobar construction, Colloque de topologie algebrique Louvain, 1956, pp. 82–87.

    Google Scholar 

  2. D.Barnes and L.A.Lambe, A fixed point approach to homological perturbation theory, Proc. AMS (to appear).

    Google Scholar 

  3. L.J. Billera, P. Filliman and B. Sturmfels, Construction and complexity of secondary polytopes, Adv. Math. 83 (1990), 155–179.

    Article  MathSciNet  MATH  Google Scholar 

  4. C.W. Lee, The associahedron and triangulations of the n-gon, Europ. J. Comb. 10 (1989), 551–560.

    Article  MathSciNet  MATH  Google Scholar 

  5. C.W. Lee, Regular triangulations of convex polytopes, preprint DIMACS Tech. Rep. 90–16.

    Google Scholar 

  6. L.C. Biedenharn, An identity satisfied by the Racah coefficients, J. Math. Phys. 31 (1953), 287–293.

    Article  MathSciNet  MATH  Google Scholar 

  7. L.C.Biedenharn and J.D.Louck, Angular Momentum in Quantum Physics, Addison-Wesley, 1981.

    Google Scholar 

  8. C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124.

    Article  MathSciNet  MATH  Google Scholar 

  9. A.Douady, Obstruction primaire à la déformation, exposé 4, Seminaire Henri CARTAN, 1960/61.

    Google Scholar 

  10. V.G. Drinfel'd, Alg. Anal. 1 no. 6 (1989), 114–148; Quasi-Hopf algebras, Leingrad Math. J. 1 (1990), 1419–1457.

    MathSciNet  Google Scholar 

  11. V.G. Drinfel'd, Alg. Anal. 1 no. 2 (1989), 30–46 (in Russian); On the concept of cocommutative Hopf algebras, Leningrad Math. J. 1 (1990).

    MathSciNet  Google Scholar 

  12. V.G. Drinfel'd, Quantum groups, Proc. ICM-86 (Berkeley), vol. 1, AMS, 1987, pp. 798–820.

    MathSciNet  MATH  Google Scholar 

  13. V.G.Drinfel'd, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, preprint ITP-89-43E (1989).

    Google Scholar 

  14. V.G. Drinfel'd, Alg. Anal. 2 no. 4 (1990), 149–181. (in Russian)

    MathSciNet  Google Scholar 

  15. S. Eilenberg and S. MacLane, On the groups H(Π, n). I, Ann. of Math. 58 (1953), 55–106.

    Article  MathSciNet  MATH  Google Scholar 

  16. J.P. Elliott, Theoretical studies in nuclear structure V, Proc. Roy. Soc. A218 (1953), 370.

    Google Scholar 

  17. B.L. Feigin, The semi-infinite homology of Kac-Moody and Virasoro Lie algebras, Russian Math. Surveys 39 (1984), 155–156; Russian original, Usp. Mat. Nauk 39 (1984), 195–196.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1964,), 59–103; On the deformation of rings and algebras III, Ann. of Math. 88 (1968), 1–34.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Gerstenhaber and S.D. Schack, Bialgebra cohomology, deformations, and quantum groups, PNAS, USA 87 (1990), 478–481.

    Article  MathSciNet  MATH  Google Scholar 

  21. M.Gerstenhaber and S.D.Schack, Algebras, bialgebras, quantum groups and algebraic deformations, Proc. Conference on Deformation Theory and Quantization with Applications to Physics, Amherst, June 1990, Contemporary Math to appear.

    Google Scholar 

  22. H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 34 (1986), 2360–2429.

    Article  MathSciNet  Google Scholar 

  23. H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 35 (1987), 1318–1355.

    Article  MathSciNet  Google Scholar 

  24. T.Kugo, String Field Theory, Lectures delivered at 25th Course of the International School of Subnuclear Physics on “The SuperWorld II”, Erice, August 6–14, (1987).

    Google Scholar 

  25. J.Huebschmann and T.Kadeishvili, Minimal models for chain algebras over a local ring, Math. Zeitschrift (to appear).

    Google Scholar 

  26. V.K.A.M. Gugenheim, On the chain complex of a fibration, Ill. J. Math. 3 (1972), 398–414.

    MathSciNet  MATH  Google Scholar 

  27. V.K.A.M. Gugenheim, On a perturbation theory for the homology of the loop-space, J. Pure & Appl. Alg. 25 (1982), 197–205.

    Article  MathSciNet  MATH  Google Scholar 

  28. V.K.A.M. Gugenheim and L. Lambe, Applications of perturbation theory in differential homological algebra I, Ill. J. Math. 33 (1989).

    Google Scholar 

  29. V.K.A.M. Gugenheim, L. Lambe and J. Stasheff, Algebraic aspects of Chen's twisting cochain, Ill. J. Math. 34 (1990), 485–502.

    MathSciNet  MATH  Google Scholar 

  30. V.K.A.M.Gugenheim, L.Lambe and J.Stasheff, Perturbation theory in differential homological algebra II, Ill. J. Math. (to appear).

    Google Scholar 

  31. V.K.A.M. Gugenheim and J. Stasheff, On perturbations and A ∞-structures, Bull. Soc. Math. de Belg. 38 (1986), 237–246.

    MathSciNet  MATH  Google Scholar 

  32. L.Lambe, Homological Perturbation Theory — Hochschild Homology and Formal Groups, Proc. Conference on Deformation Theory and Quantization with Applications to Physics, Amherst, June 1990, AMS, to appear.

    Google Scholar 

  33. L. Lambe and J.D. Stasheff, Applications of perturbation theory to iterated fibrations, Manuscripta Math. 58 (1987), 363–376.

    Article  MathSciNet  MATH  Google Scholar 

  34. E.Jones, A study of Lie and associative algebras from a homotopy point of view, Master's Project, NCSU (1990).

    Google Scholar 

  35. M. Kaku, Why are there two BRST string field theories?, Phys. Lett. B 200 (1988), 22–30.

    Article  MathSciNet  Google Scholar 

  36. M.Kauku, Deriving the four-string interaction from geometric string field theory, preprint, CCNY-HEP88/5.

    Google Scholar 

  37. M.Kaku, Geometric derivation of string field theory from first principles: Closed strings and modular invariance, preprint, CCNY-HEP-88/6.

    Google Scholar 

  38. M.Kaku, Introduction to Superstrings, Springer-Verlag, 1988.

    Google Scholar 

  39. M.Kaku and J.Lykken, Modular invariant closed string field theory, preprint, CCNY-HEP-88/7.

    Google Scholar 

  40. T. Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37.4 (1987), 139–160.

    Article  MathSciNet  MATH  Google Scholar 

  41. T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Proc. Conf. on Artin's Braid Groups, Santa Cruz, 1986, vol. 78, AMS, 1988, pp. 339–363.

    MathSciNet  Google Scholar 

  42. T.Kohno, Quantized universal enveloping algebras and monodromy of braid groups, Nagoya preprint (1988).

    Google Scholar 

  43. T. Kugo, H. Kunitomo and K. Suehiro, Non-polynomial closed string field theory, Phys. Lett. 226B (1989), 48–54.

    Article  MathSciNet  Google Scholar 

  44. T. Kugo and K. Suehiro, Nonpolynomial closed string field theory: Action and gauge invariance, Nucl. Phys. B 337 (1990), 434–466.

    Article  MathSciNet  Google Scholar 

  45. R. Lashof, Classification of fibre bundles by the loop space of the base, Annals of Math. 64 (1956), 436–446.

    Article  MathSciNet  MATH  Google Scholar 

  46. P.A.B. Lecomte, P.W. Michor and H. Schicketanz, The multigraded Nijenhuis-Richardson algebra, its universal property and applications, JPAA (to appear).

    Google Scholar 

  47. P.A.B. Lecomte and C. Roger, Modules et cohomologies des bigèbres de Lie, C.R.A.S., Paris 310 (1990), 405–410.

    MathSciNet  MATH  Google Scholar 

  48. S.MacLane, Categories for the working mathematician, Springer-Verlag, 1971.

    Google Scholar 

  49. S. MacLane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), 28–46.

    MathSciNet  MATH  Google Scholar 

  50. S. Majid, Quasitriangular Hopf algebras and Yang-Baxter equations, Int. J. Mod. Phys. A 5 (1990), 1–91.

    Article  MathSciNet  MATH  Google Scholar 

  51. M.Markl, A cohomology theory for A(m)-algebras, JPAA volume in honour of Alex Heller.

    Google Scholar 

  52. W. Michaelis, Lie coalgebras, Adv. in Math. 38 (1980), 1–54.

    Article  MathSciNet  MATH  Google Scholar 

  53. J.C. Moore, The double suspension and p-primary components of the homotopy groups of spheres, Bol. Soc. Math. Mex. 1 (1956), 28–37.

    MathSciNet  MATH  Google Scholar 

  54. V.S.Retakh, Lie-Massey brackets and n-homotopically multiplicative maps of DG-Lie algebras, JPAA volume in honour of Alex Heller.

    Google Scholar 

  55. V.S. Retakh, Massey operations in the cohomology of Lie superalgebras and deformations of complex analytical algebras, Funct. Anal. Appl 11 no. 4 (1977), 88–89.

    MathSciNet  Google Scholar 

  56. V.S. Retakh, Massey operations in Lie superalgebras and differentials in Quillen spectral sequences, Funct. Anal. Appl 12 no. 4 (1978), 91–92.

    MathSciNet  Google Scholar 

  57. V.S. Retakh, Massey operations in Lie superalgebras and differentials in Quillen spectral sequences, Colloquim Mathematicum no. 50 (1985), 81–94. (Russian)

    Google Scholar 

  58. M. Saadi and B. Zwiebach, Closed string field theory from polyhedra, Ann. Phys. (N.Y.) 192 (1989), 213–227.

    Article  MathSciNet  Google Scholar 

  59. M. Schlessinger and J. D. Stasheff, The Lie algebra structure of tangent cohomology and deformation theory, J. of Pure and Appl. Algebra 38 (1985), 313–322.

    Article  MathSciNet  MATH  Google Scholar 

  60. M. Schlessinger and J. D. Stasheff, Deformation theory and rational homotopy type, Publ. Math. IHES (to appear — eventually).

    Google Scholar 

  61. J. Stasheff, An almost groupoid structure for the space of (open) strings and implications for string field theory, Advances in Homotopy Theory, LMS Lect. Note Series 139, 1989, pp. 165–172.

    Google Scholar 

  62. J. Stasheff, Drinfel'd's quasi-Hopf algebras and beyond, Proc. Conference on Deformation Theory and Quantization with Applications to Physics, Amherst, June 1990, AMS, to appear.

    Google Scholar 

  63. J.D. Stasheff, H-spaces from a homotopy point of view, LNM 161, Springer-Verlag, 1970.

    Google Scholar 

  64. J.D. Stasheff, On the homotopy associativity of H-spaces I, Trans. AMS 108 (1963), 275–292.

    Article  MathSciNet  MATH  Google Scholar 

  65. J.D. Stasheff, On the homotopy associativity of H-spaces II, Trans. AMS 108 (1963), 293–312.

    MathSciNet  MATH  Google Scholar 

  66. J. Stasheff, The intrinsic bracket on the deformation complex of an associative algebra, JPAA volume in honour of Alex Heller.

    Google Scholar 

  67. D. Sullivan, Infinitesimal computations in topology, Publ. Math. IHES 47 (1977), 269–331.

    Article  MathSciNet  MATH  Google Scholar 

  68. H.-W. Wiesbrock, A note on the construction of the C*-Algebra of bosonic strings, preprint, FUB-HEP-90/.

    Google Scholar 

  69. H.-W. Wiesbrock, The quantum algebra of bosonic strings, preprint, FUB-HEP/89-9.

    Google Scholar 

  70. H.-W. Wiesbrock, The mathematics of the string algebra, preprint, DESY 90-003.

    Google Scholar 

  71. E. Witten, Non-commutative geometry and string field theory, Nuclear Physics B 268 (1986), 253–294.

    Article  MathSciNet  Google Scholar 

  72. E. Witten, Interacting field theory of open strings, Nuclear Physics B 276, (1986), 291–324.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, University of North Carolina at Chapel Hill, CB# 3250, 27599-3250, Phillips Hall Chapel Hill, NC, USA

    Jim Stasheff

Authors
  1. Jim Stasheff
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Petr P. Kulish

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Stasheff, J. (1992). Differential graded Lie algebras, quasi-hopf algebras and higher homotopy algebras. In: Kulish, P.P. (eds) Quantum Groups. Lecture Notes in Mathematics, vol 1510. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101184

Download citation

  • DOI: https://doi.org/10.1007/BFb0101184

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55305-2

  • Online ISBN: 978-3-540-47020-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation