Advertisement

An introduction to Tannaka duality and quantum groups

  • André Joyal
  • Ross Street
Part II
Part of the Lecture Notes in Mathematics book series (LNM, volume 1488)

Keywords

Hopf Algebra Quantum Group Compact Group Natural Transformation Monoidal Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AK]
    R.A. Askey, T.H. Koornwinder and W. Schempp, Special Functions: Group Theoretical Aspects and Applications, Mathematics and its Applications (D. Reidel, 1984).Google Scholar
  2. [BL1]
    L.C. Biedenharm and J.D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and its Applications 8 (Addison-Wesley, 1981).Google Scholar
  3. [BL2]
    L.C. Biedenharm and J.D. Louck, The Racah-Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics and its Applications 9 (Addison-Wesley, 1982).Google Scholar
  4. [BtD]
    T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Math. 98 (Springer-Verlag, Berlin 1985).zbMATHGoogle Scholar
  5. [Car]
    P. Cartier, Development recents sur les groupes de tresses. Applications à la topologie et à l'algèbre, Seminaire Bourbaki 42 ie annee, 1989–90, n o 716.Google Scholar
  6. [C]
    C. Chevalley, Theory of Lie Groups, (Princeton University Press, 1946).Google Scholar
  7. [DM]
    P. Deligne and J.S. Milne, Tannakian categories, Hodge Cocycles Motives and Shimura Varieties, Lecture Notes in Math. 900 (Springer-Verlag, Berlin 1982) 101–228.CrossRefGoogle Scholar
  8. [DHR]
    S. Doplicher, R. Haag, J.E. Roberts, Local observables and particle statistics II, Comm. Math. Phys. 35 (1974) 49–85.MathSciNetCrossRefGoogle Scholar
  9. [DR]
    S. Doplicher and J.E. Roberts, A new duality theory for compact groups, Inventiones Math. 98 (1989) 157–218.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [D1]
    V.G. Drinfel'd, Quantum groups, Proceedings of the International Congress of Mathematicians at Berkeley, California, U.S.A. 1986 (1987) 798–820.Google Scholar
  11. [D2]
    V.G. Drinfel'd, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, Acad. Sci. Ukr. (Preprint, ITP-89-43E, 1989)Google Scholar
  12. [FRT]
    L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan, Quantization of Lie algebras and Lie groups, LOMI Preprints (Leningrad 1987); Algebra i Analiz 1:1 (1989, in Russian).Google Scholar
  13. [F]
    P. Freyd, Abelian Categories, (Harper & Row, New York 1964).zbMATHGoogle Scholar
  14. [FY]
    P. Freyd and D. Yetter, Braided compact closed categories with applications to low dimensional topology, Advances in Math. 77 (1989) 156–182.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [IS]
    N. Iwahori and M. Sugiura, A duality theorem for homogeneous compact manifolds of compact Lie groups, Osaka J. Math. 3 (1966) 139–153.MathSciNetzbMATHGoogle Scholar
  16. [J]
    M. Jimbo, A q-difference analog of U(g) and the Yang-Baxter equation, Lett. Math. Phys.10 (1985) 63–69.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Jn1]
    V. Jones, A polynomial invariant for knots via von Neumann algebras, Bulletin American Math. Soc. 12 (1985) 103–111.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [Jn2]
    V. Jones, Index for subfactors, Inventiones Math. 72 (1983) 1–25.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [JS1]
    A. Joyal and R. Street, Braided monoidal categories, Macquarie Math. Reports #850067(Dec. 1985); Revised #860081 (Nov. 1986).Google Scholar
  20. [JS2]
    A. Joyal and R. Street, The geometry of tensor calculus I, Advances in Math. (to appear).Google Scholar
  21. [JS3]
    A. Joyal and R. Street, Braided tensor categories, Advances in Math. (to appear).Google Scholar
  22. [JS4]
    A. Joyal and R. Street, Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991) 43–51.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [K]
    G.M. Kelly, On clubs and doctrines, Category Seminar Sydney 1972–73, Lecture Notes in Math. 420 (Springer-Verlag, Berlin 1974) 181–256.Google Scholar
  24. [K1]
    S. Kleinman, Motives, Proceedings of the Fifth Nordic Summer School, Oslo 1970 (Wolters-Noordhoff, Holland 1972).Google Scholar
  25. [Ko]
    T.H. Koornwinder, Orthogonal polynomials in connection with quantum groups.Google Scholar
  26. [Kr]
    M.G. Krein, A principle of duality for a bicompact group and square block algebra, Dokl. Akad. Nauk. SSSR 69 (1949) 725–728.MathSciNetGoogle Scholar
  27. [Le]
    G. Lewis, Coherence for a closed functor, Coherence in Categories, Lecture Notes in Math. 281 (Springer-Verlag, Berlin 1972) 148–195.CrossRefGoogle Scholar
  28. [Lu]
    G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Advances in Math. 70 (1988) 237–249.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [Ly]
    V.V. Lyubashenko, Hopf algebras and vector symmetries, Uspekhi Mat. Nauk. 41 #5 (1986) 185–186.MathSciNetzbMATHGoogle Scholar
  30. [ML]
    S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Math. 5 (Springer-Verlag, Berlin 1971).CrossRefzbMATHGoogle Scholar
  31. [Mn]
    Yu.I. Manin, Quantum Groups and Non-Commutative Geometry, Les publications du Centre de Recherches Mathématiques (Université de Montréal, 4e trimestre 1988).Google Scholar
  32. [M1]
    S. Majid, Representations, duals and quantum doubles of monoidal categories, Supl. Rend. Circ. Mat. Palermo (to appear).Google Scholar
  33. [M2]
    S. Majid, Braided groups, Preprint, DAMTP/92-42 (Cambridge 1990).Google Scholar
  34. [M3]
    S. Majid, Tannaka-Krein theorem for quasiHopf algebras and other results, Contemp. Math. (to appear).Google Scholar
  35. [P]
    B. Pareigis, A non-commutative non-cocommutative Hopf algebra in nature, J. Algebra 70 (1981) 356–374.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [Pn]
    R. Penrose, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications, Edited by D.J.A. Welsh (Academic Press, 1971) 221–244.Google Scholar
  37. [Pd]
    P. Podles, Quantum spheres, Lett. Math. Phys. (1987)193–202.Google Scholar
  38. [RT]
    N.Yu. Reshetikhin and V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127(1) (1990) 1–26.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [R]
    M. Rosso, C.R. Acad. Sc. Paris 305, Série I (1987) 587–590.Google Scholar
  40. [SR]
    N. Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Math. 265 (Springer-Verlag, Berlin 1972).CrossRefzbMATHGoogle Scholar
  41. [Sc]
    L. Schwartz, Théorie des Distributions, (Hermann, Paris 1966).zbMATHGoogle Scholar
  42. [Sh]
    M.C. Shum, Tortile Tensor Categories, PhD Thesis (Macquarie University, November 1989); Macquarie Math. Reports #900047(Apr. 1990).Google Scholar
  43. [Sw]
    M.E. Sweedler, Hopf Algebras, Mathematical Lecture Notes Series (Benjamin, 1969).Google Scholar
  44. [Ta]
    T Tannaka, Über den Dualitätssatz der nichtkommutativen topologischen Gruppen, Tôhoku Math. J. 45 (1939) 1–12.zbMATHGoogle Scholar
  45. [T]
    V.G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988) 527–553.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [U]
    K.-H. Ulbrich, On Hopf algebras and rigid monoidal categories, Israel J. Math 70 (to appear).Google Scholar
  47. [W]
    S.L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Inventiones Math. 93 (1988) 35–76.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [Y1]
    D.N. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Camb. Phil. Soc. (to appear).Google Scholar
  49. [Y2]
    D.N. Yetter, Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories, Preprint.Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • André Joyal
    • 1
    • 2
  • Ross Street
    • 1
    • 2
  1. 1.Université du Québec à MontréalCanada
  2. 2.Macquarie UniversityAustralia

Personalised recommendations