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Tensor Products of Modules for a Vertex Operator Algebra and Vertex Tensor Categories

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Lie Theory and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 123))

Abstract

In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announcement has also appeared [HL1]. The theory is based on both the formal-calculus approach to vertex operator algebra theory developed in [FLM2] and [FHL] and the precise geometric interpretation of the notion of vertex operator algebra established in [H1].

Dedicated to Bert Kostant on the occasion of his sixty-fifth birthday

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References

  1. A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetries in two-dimensional quantum field theory, Nucl. Phys. B241 (1984), 333–380.

    Article  MathSciNet  Google Scholar 

  2. R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068–3071.

    Article  MathSciNet  Google Scholar 

  3. R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent Math. 109 (1992), 405–444.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Brustein, Y. Ne’eman and S. Sternberg, Duality, crossing and Mac Lane’s coherence, Isr. J. Math. 72 (1990), 19–37.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308–339.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Dong, Representations of the moonshine module vertex operator algebra, in: Proc. 1992 Joint Summer Research Conference on Conformai Field Theory, Topological Field Theory and Quantum Groups, Mount Holyoke, 1992, Contemporary Math., Amer. Math. Soc., Providence, to appear.

    Google Scholar 

  7. C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progress in Math., Vol. 112, Birkhäuser, Boston, 1993.

    Book  MATH  Google Scholar 

  8. V. Drinfeld, On quasi-cocommutative Hopf algebras, Algebra and Analysis 1 (1989), 30–46.

    MathSciNet  Google Scholar 

  9. V. Drinfeld, On quasitriangular quasi-Hopf algebras and a certain group closely related to Gal(<Inline>1</Inline>/ℚ), Algebra and Analysis 2 (1990), 149–181.

    MathSciNet  Google Scholar 

  10. G. Faltings, A proof for the Verlinde formula, to appear.

    Google Scholar 

  11. M. Finkelberg, Fusion categories, Ph.D. thesis, Harvard University, 1993.

    Google Scholar 

  12. I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, preprint, 1989; Memoirs Amer. Math. Soc. 104, Number 494, 1993.

    MathSciNet  Google Scholar 

  13. I. B. Frenkel, J. Lepowsky and A. Meurman, A natural representation of the Fischer-Griess Monster with the modular function J as character, Proc. Natl. Acad. Sci. USA 81 (1984), 3256–3260.

    Article  MathSciNet  MATH  Google Scholar 

  14. I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988.

    MATH  Google Scholar 

  15. I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123–168.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Friedan and S. Shenker, The analytic geometry of two-dimensional conformal field theory, Nucl. Phys. B281 (1987), 509–545.

    Article  MathSciNet  Google Scholar 

  17. D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum Press, New York, 1982.

    MATH  Google Scholar 

  18. R. L. Griess, Jr., The Friendly Giant, Invent. Math. 69 (1982), 1–102.

    Article  MathSciNet  MATH  Google Scholar 

  19. Y.-Z. Huang, On the geometric interpretation of vertex operator algebras, Ph.D. thesis, Rutgers University, 1990; Operads and the geometric interpretation of vertex operator algebras, I, to appear.

    Google Scholar 

  20. Y.-Z. Huang, Geometric interpretation of vertex operator algebras, Proc. Natl. Acad. Sci. USA 88 (1991), 9964–9968.

    Article  MATH  Google Scholar 

  21. Y.-Z. Huang, Applications of the geometric interpretation of vertex operator algebras, in: Proc. 20th International Conference on Differential Geometric Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scientific, Singapore, 1992, Vol. 1, 333–343.

    Google Scholar 

  22. Y.-Z. Huang, Vertex operator algebras and conformal field theory, Intl. J. Mod. Phys. A7 (1992), 2109–2151.

    Article  Google Scholar 

  23. Y.-Z. Huang, Operads and the geometric interpretation of vertex operator algebras, II, in preparation.

    Google Scholar 

  24. Y.-Z. Huang and J. Lepowsky, Toward a theory of tensor products for representations of a vertex operator algebra, in: Proc. 20th International Conference on Differential Geometric Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scientific, Singapore, 1992, Vol. 1, 344–354.

    Google Scholar 

  25. Y.-Z. Huang and J. Lepowsky, Vertex operator algebras and operads, in: The Gelfand Mathematical Seminars, 1990–1992, ed. L. Corwin, I. Gelfand and J. Lepowsky, Birkhäuser, Boston, 1993, 145–161.

    Chapter  Google Scholar 

  26. Y.-Z. Huang and J. Lepowsky, Operadic formulation of the notion of vertex operator algebra, in: Proc. 1992 Joint Summer Research Conference on Conformal Field Theory, Topological Field Theory and Quantum Groups, Mount Holyoke, 1992, Contemporary Math., Amer. Math. Soc, Providence, to appear.

    Google Scholar 

  27. Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, I, to appear.

    Google Scholar 

  28. Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, II, to appear.

    Google Scholar 

  29. V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987), 335–388.

    Article  MATH  Google Scholar 

  30. A. Joyal and R. Street, Braided monoidal categories, Macquarie Mathematics Reports, Macquarie University, Australia, 1986.

    Google Scholar 

  31. A. Joyal and R. Street, An introduction to Tannaka duality and quantum groups, in: Category theory, Como, 1990, ed. A. Carboni, M.C. Pedicchio and G. Rosolini, Lecture Notes in Math. 1488, Springer-Verlag, Berlin, 1991, 413–492.

    Google Scholar 

  32. D. Kazhdan and G. Lusztig, Affine Lie algebras and quantum groups, International Mathematics Research Notices (in Duke Math. J.) 2 (1991), 21–29.

    Article  MathSciNet  Google Scholar 

  33. D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, I, J. Amer. Math. Soc. 6 (1993), 905–947.

    Article  MathSciNet  MATH  Google Scholar 

  34. D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, II, J. Amer. Math. Soc. 6 (1993), 949–1011.

    Article  MathSciNet  Google Scholar 

  35. D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, III, J. Amer. Math. Soc. 7 (1994), 335–381.

    Article  MathSciNet  MATH  Google Scholar 

  36. D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, IV, J. Amer. Math. Soc. 7 (1994), 383–453.

    Article  MathSciNet  MATH  Google Scholar 

  37. V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino models in two dimensions, Nucl. Phys. B247 (1984), 83–103.

    Article  MathSciNet  Google Scholar 

  38. T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, in: Braids, Santa Cruz, 1986, Contemporary Math. 78 (1988), 339–363.

    Google Scholar 

  39. T. Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37 (1987), 139–160.

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Lepowsky, Perspectives on vertex operators and the Monster, in: Proc. 1987 Symposium on the Mathematical Heritage of Hermann Weyl, Duke Univ., Proc. Symp. Pure Math., Amer. Math. Soc. 48 (1988), 181–197.

    Google Scholar 

  41. J. Lepowsky, Remarks on vertex operator algebras and moonshine, in: Proc. 20th International Conference on Differential Geometric Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scientific, Singapore, 1992, Vol. 1, 362–370.

    Google Scholar 

  42. J. Lepowsky and R.L. Wilson, A new family of algebras underlying the Rogers-Ramanujan identities and generalizations, Proc. Natl. Acad. Sci. USA 78 (1981), 7254–7258.

    Article  MathSciNet  MATH  Google Scholar 

  43. S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Math., Vol. 5, Springer-Verlag, New York, 1971.

    Book  MATH  Google Scholar 

  44. G. Mason, Finite groups and modular functions (with an appendix by S. P. Norton), in: Representations of Finite Groups, Proc. 1986 Summer Research Institute, Areata, ed. P. Fong, Proc. Symp. Pure Math., Amer. Math. Soc. 47 (1987), 181–210.

    Google Scholar 

  45. J. P. May, The geometry of iterated loop spaces, Lecture Notes in Math. 271, Springer-Verlag, 1972.

    Google Scholar 

  46. G. Moore and N. Seiberg, Classical and quantum Conformal field theory, Comm. Math. Phys. 123 (1989), 177–254.

    Article  MathSciNet  MATH  Google Scholar 

  47. N. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547–597.

    Article  MathSciNet  MATH  Google Scholar 

  48. V. V. Schechtman and A. N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent Math. 106 (1991), 139–194.

    Article  MathSciNet  MATH  Google Scholar 

  49. J. D. Stasheff, Homotopy associativity of H-spaces, I, Trans. Amer. Math. Soc. 108 (1963), 275–292.

    Article  MathSciNet  MATH  Google Scholar 

  50. J. D. Stasheff, Homotopy associativity of H-spaces, II, Trans. Amer. Math. Soc. 108 (1963), 293–312.

    MathSciNet  Google Scholar 

  51. A. Tsuchiya and Y. Kanie, Vertex operators in Conformal field theory on ℙ1 and monodromy representations of braid group, in: Conformal Field Theory and Solvable Lattice Models, Advanced Studies in Pure Math., Vol. 16, Kinokuniya Company Ltd., Tokyo, 1988, 297–372.

    Google Scholar 

  52. A. Tsuchiya, K. Ueno and Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, in: Advanced Studies in Pure Math., Vol. 19, Kinokuniya Company Ltd., Tokyo, 1989, 459–565.

    Google Scholar 

  53. A. N. Varchenko, Hypergeometric functions and representation theory of Lie algebras and quantum groups, to appear.

    Google Scholar 

  54. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B300 (1988), 360–376.

    Article  MathSciNet  Google Scholar 

  55. W. Wang, Rationality of Virasoro vertex operator algebras, International Mathematics Research Notices (in Duke Math. J.) 7 (1993), 197–211.

    Article  Google Scholar 

  56. E. Witten, Non-abelian bosonization in two dimensions, Comm. Math. Phys. 92 (1984), 455–472.

    Article  MathSciNet  MATH  Google Scholar 

  57. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351–399

    Article  MathSciNet  MATH  Google Scholar 

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Author notes

  1. Yi-Zhi Huang

    Present address: Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Yi-Zhi Huang

  2. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    James Lepowsky

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  1. Yi-Zhi Huang
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  2. James Lepowsky
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  1. Department of Mathematics, Penn State University, 16802, University Park, PA, USA

    Jean-Luc Brylinski  & Ranee Brylinski  & 

  2. Department of Mathematics, MIT, 02139, Cambridge, MA, USA

    Victor Guillemin  & Victor Kac  & 

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Huang, YZ., Lepowsky, J. (1994). Tensor Products of Modules for a Vertex Operator Algebra and Vertex Tensor Categories. In: Brylinski, JL., Brylinski, R., Guillemin, V., Kac, V. (eds) Lie Theory and Geometry. Progress in Mathematics, vol 123. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0261-5_13

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  • DOI: https://doi.org/10.1007/978-1-4612-0261-5_13

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6685-3

  • Online ISBN: 978-1-4612-0261-5

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