Quantum Groups pp 120-137 | Cite as

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

  • Jim Stasheff
I. Quantum Groups, Deformation Theory And Representation Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1510)

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References

  1. [A]
    J.F.Adams, On the cobar construction, Colloque de topologie algebrique Louvain, 1956, pp. 82–87.Google Scholar
  2. [BL]
    D.Barnes and L.A.Lambe, A fixed point approach to homological perturbation theory, Proc. AMS (to appear).Google Scholar
  3. [Ass]a
    L.J. Billera, P. Filliman and B. Sturmfels, Construction and complexity of secondary polytopes, Adv. Math. 83 (1990), 155–179.MathSciNetCrossRefMATHGoogle Scholar
  4. [Ass]b
    C.W. Lee, The associahedron and triangulations of the n-gon, Europ. J. Comb. 10 (1989), 551–560.MathSciNetCrossRefMATHGoogle Scholar
  5. [Ass]c
    C.W. Lee, Regular triangulations of convex polytopes, preprint DIMACS Tech. Rep. 90–16.Google Scholar
  6. [B]
    L.C. Biedenharn, An identity satisfied by the Racah coefficients, J. Math. Phys. 31 (1953), 287–293.MathSciNetCrossRefMATHGoogle Scholar
  7. [BL]
    L.C.Biedenharn and J.D.Louck, Angular Momentum in Quantum Physics, Addison-Wesley, 1981.Google Scholar
  8. [CE]
    C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124.MathSciNetCrossRefMATHGoogle Scholar
  9. [Dou]
    A.Douady, Obstruction primaire à la déformation, exposé 4, Seminaire Henri CARTAN, 1960/61.Google Scholar
  10. [D1]
    V.G. Drinfel'd, Alg. Anal. 1 no. 6 (1989), 114–148; Quasi-Hopf algebras, Leingrad Math. J. 1 (1990), 1419–1457.MathSciNetGoogle Scholar
  11. [D2]
    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).MathSciNetGoogle Scholar
  12. [D3]
    V.G. Drinfel'd, Quantum groups, Proc. ICM-86 (Berkeley), vol. 1, AMS, 1987, pp. 798–820.MathSciNetMATHGoogle Scholar
  13. [D4]
    V.G.Drinfel'd, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, preprint ITP-89-43E (1989).Google Scholar
  14. [D5]
    V.G. Drinfel'd, Alg. Anal. 2 no. 4 (1990), 149–181. (in Russian)MathSciNetGoogle Scholar
  15. [EM]
    S. Eilenberg and S. MacLane, On the groups H(Π, n). I, Ann. of Math. 58 (1953), 55–106.MathSciNetCrossRefMATHGoogle Scholar
  16. [E]
    J.P. Elliott, Theoretical studies in nuclear structure V, Proc. Roy. Soc. A218 (1953), 370.Google Scholar
  17. [F]
    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.MathSciNetCrossRefMATHGoogle Scholar
  18. [G1]
    M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. 78 (1963), 267–288.MathSciNetCrossRefMATHGoogle Scholar
  19. [G2]
    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.MathSciNetCrossRefMATHGoogle Scholar
  20. [GS]a
    M. Gerstenhaber and S.D. Schack, Bialgebra cohomology, deformations, and quantum groups, PNAS, USA 87 (1990), 478–481.MathSciNetCrossRefMATHGoogle Scholar
  21. [GS]b
    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. [HIKKO]a
    H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 34 (1986), 2360–2429.MathSciNetCrossRefGoogle Scholar
  23. [HIKKO]b
    H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 35 (1987), 1318–1355.MathSciNetCrossRefGoogle Scholar
  24. [HIKKO]c
    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. [HK]
    J.Huebschmann and T.Kadeishvili, Minimal models for chain algebras over a local ring, Math. Zeitschrift (to appear).Google Scholar
  26. [HPT]a
    V.K.A.M. Gugenheim, On the chain complex of a fibration, Ill. J. Math. 3 (1972), 398–414.MathSciNetMATHGoogle Scholar
  27. [HPT]b
    V.K.A.M. Gugenheim, On a perturbation theory for the homology of the loop-space, J. Pure & Appl. Alg. 25 (1982), 197–205.MathSciNetCrossRefMATHGoogle Scholar
  28. [HPT]c
    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. [HPT]d
    V.K.A.M. Gugenheim, L. Lambe and J. Stasheff, Algebraic aspects of Chen's twisting cochain, Ill. J. Math. 34 (1990), 485–502.MathSciNetMATHGoogle Scholar
  30. [HPT]e
    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. [HPT]f
    V.K.A.M. Gugenheim and J. Stasheff, On perturbations and A -structures, Bull. Soc. Math. de Belg. 38 (1986), 237–246.MathSciNetMATHGoogle Scholar
  32. [HPT]j
    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. [HPT]h
    L. Lambe and J.D. Stasheff, Applications of perturbation theory to iterated fibrations, Manuscripta Math. 58 (1987), 363–376.MathSciNetCrossRefMATHGoogle Scholar
  34. [J]
    E.Jones, A study of Lie and associative algebras from a homotopy point of view, Master's Project, NCSU (1990).Google Scholar
  35. [K]a
    M. Kaku, Why are there two BRST string field theories?, Phys. Lett. B 200 (1988), 22–30.MathSciNetCrossRefGoogle Scholar
  36. [K]b
    M.Kauku, Deriving the four-string interaction from geometric string field theory, preprint, CCNY-HEP88/5.Google Scholar
  37. [K]c
    M.Kaku, Geometric derivation of string field theory from first principles: Closed strings and modular invariance, preprint, CCNY-HEP-88/6.Google Scholar
  38. [K]d
    M.Kaku, Introduction to Superstrings, Springer-Verlag, 1988.Google Scholar
  39. [K]e
    M.Kaku and J.Lykken, Modular invariant closed string field theory, preprint, CCNY-HEP-88/7.Google Scholar
  40. [Ko]a
    T. Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37.4 (1987), 139–160.MathSciNetCrossRefMATHGoogle Scholar
  41. [Ko]b
    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.MathSciNetGoogle Scholar
  42. [Ko]c
    T.Kohno, Quantized universal enveloping algebras and monodromy of braid groups, Nagoya preprint (1988).Google Scholar
  43. [KKS]a
    T. Kugo, H. Kunitomo and K. Suehiro, Non-polynomial closed string field theory, Phys. Lett. 226B (1989), 48–54.MathSciNetCrossRefGoogle Scholar
  44. [KKS]b
    T. Kugo and K. Suehiro, Nonpolynomial closed string field theory: Action and gauge invariance, Nucl. Phys. B 337 (1990), 434–466.MathSciNetCrossRefGoogle Scholar
  45. [L]
    R. Lashof, Classification of fibre bundles by the loop space of the base, Annals of Math. 64 (1956), 436–446.MathSciNetCrossRefMATHGoogle Scholar
  46. [LMS]
    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. [LR]
    P.A.B. Lecomte and C. Roger, Modules et cohomologies des bigèbres de Lie, C.R.A.S., Paris 310 (1990), 405–410.MathSciNetMATHGoogle Scholar
  48. [MacL]a
    S.MacLane, Categories for the working mathematician, Springer-Verlag, 1971.Google Scholar
  49. [MacL]b
    S. MacLane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), 28–46.MathSciNetMATHGoogle Scholar
  50. [Maj]
    S. Majid, Quasitriangular Hopf algebras and Yang-Baxter equations, Int. J. Mod. Phys. A 5 (1990), 1–91.MathSciNetCrossRefMATHGoogle Scholar
  51. [Mar]
    M.Markl, A cohomology theory for A(m)-algebras, JPAA volume in honour of Alex Heller.Google Scholar
  52. [Mi]
    W. Michaelis, Lie coalgebras, Adv. in Math. 38 (1980), 1–54.MathSciNetCrossRefMATHGoogle Scholar
  53. [M]
    J.C. Moore, The double suspension and p-primary components of the homotopy groups of spheres, Bol. Soc. Math. Mex. 1 (1956), 28–37.MathSciNetMATHGoogle Scholar
  54. [R]a
    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. [R]b
    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.MathSciNetGoogle Scholar
  56. [R]c
    V.S. Retakh, Massey operations in Lie superalgebras and differentials in Quillen spectral sequences, Funct. Anal. Appl 12 no. 4 (1978), 91–92.MathSciNetGoogle Scholar
  57. [R]d
    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. [SZ]
    M. Saadi and B. Zwiebach, Closed string field theory from polyhedra, Ann. Phys. (N.Y.) 192 (1989), 213–227.MathSciNetCrossRefGoogle Scholar
  59. [SS]a
    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.MathSciNetCrossRefMATHGoogle Scholar
  60. [SS]b
    M. Schlessinger and J. D. Stasheff, Deformation theory and rational homotopy type, Publ. Math. IHES (to appear — eventually).Google Scholar
  61. [S1]
    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. [S2]
    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. [S3]
    J.D. Stasheff, H-spaces from a homotopy point of view, LNM 161, Springer-Verlag, 1970.Google Scholar
  64. [S4]
    J.D. Stasheff, On the homotopy associativity of H-spaces I, Trans. AMS 108 (1963), 275–292.MathSciNetCrossRefMATHGoogle Scholar
  65. [S5]
    J.D. Stasheff, On the homotopy associativity of H-spaces II, Trans. AMS 108 (1963), 293–312.MathSciNetMATHGoogle Scholar
  66. [S6]
    J. Stasheff, The intrinsic bracket on the deformation complex of an associative algebra, JPAA volume in honour of Alex Heller.Google Scholar
  67. [Su]
    D. Sullivan, Infinitesimal computations in topology, Publ. Math. IHES 47 (1977), 269–331.MathSciNetCrossRefMATHGoogle Scholar
  68. [Wies]a
    H.-W. Wiesbrock, A note on the construction of the C*-Algebra of bosonic strings, preprint, FUB-HEP-90/.Google Scholar
  69. [Wies]b
    H.-W. Wiesbrock, The quantum algebra of bosonic strings, preprint, FUB-HEP/89-9.Google Scholar
  70. [Wies]c
    H.-W. Wiesbrock, The mathematics of the string algebra, preprint, DESY 90-003.Google Scholar
  71. [W]a
    E. Witten, Non-commutative geometry and string field theory, Nuclear Physics B 268 (1986), 253–294.MathSciNetCrossRefGoogle Scholar
  72. [W]b
    E. Witten, Interacting field theory of open strings, Nuclear Physics B 276, (1986), 291–324.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Jim Stasheff
    • 1
  1. 1.Mathematics DepartmentUniversity of North Carolina at Chapel HillPhillips Hall Chapel HillUSA

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