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Generalized trace and modified dimension functions on ribbon categories

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Abstract

In this paper, we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are nonzero. Such examples include categories of finite dimensional modules of certain Lie algebras and finite groups over a field of positive characteristic and categories of finite dimensional modules of basic Lie superalgebras over the complex numbers. These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory.

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References

  1. Alperin J.L.: Local Representation Theory. Studies in Advanced Mathematics, 11. Cambridge University Press, Cambridge (1986)

    Book  Google Scholar 

  2. Andersen H.H.: Tensor products of quantized tilting modules. Comm. Math. Phys. 149, 149–159 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Archer L.: On certain quotients of the Green rings of dihedral 2-groups. J. Pure Appl. Algebra 212(8), 1888–1897 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bakalov B., Kirillov A. Jr.: Lectures on Tensor Categories and Modular Functors. University Lecture Series, 21. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  5. Balmer P.: Supports and filtrations in algebraic geometry and modular representation theory. Amer. J. Math. 129(5), 1227–1250 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bar-Natan D., Le T., Thurston D.: Two applications of elementary knot theory to Lie algebras and Vassiliev invariants. Geom. Topol. 7, 1–31 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bas̆ev V.A.: Representations of the group Z 2 × Z 2 in a field of characteristic 2, (Russian). Dokl. Akad. Nauk SSSR 141, 1015–1018 (1961)

    MathSciNet  Google Scholar 

  8. Benkart, G., Osborn, M.: Representations of Rank one Lie Algebras of Characteristic p, Lie Algebras and Related Topics, pp. 1–37, Lecture Notes in Math., p. 933. Springer, Berlin-New York (1982)

  9. Benson D.J.: Representations and Cohomology, I: Basic representation theory of finite groups and associative algebras. Second edition. Cambridge Studies in Advanced Mathematics, 30. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  10. Benson D.J.: Representations and Cohomology, II: Cohomology of groups and modules. Second edition. Cambridge Studies in Advanced Mathematics, 31. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  11. Benson D.J., Carlson J.F.: Nilpotent elements in the Green ring. J. Algebra 104(2), 329–350 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. Berele A., Regev A.: Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras. Adv. Math. 64, 118–175 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Boe, B., Kujawa, J., Nakano, D.: Cohomology and support varieties for Lie superalgebras, Preprint.

  14. Boe, B., Kujawa, J., Nakano, D.: Cohomology and support varieties for Lie superalgebras, II, Proceedings of LMS, to appear.

  15. Brundan J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra \({{\mathfrak{gl}(m|n)}}\). J. Amer. Math. Soc. 16, 185–231 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carlson J.F., Peng C.: Relative projectivity and ideals in cohomology rings. J. Algebra 183(3), 929–948 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Conlon S.B.: Certain representation algebras. J. Aust. Math. Soc. 5, 83–99 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  18. Deligne, P.: La catégorie des représentations du groupe symétrique S t , lorsque t n’est pas un entier naturel, Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., pp. 209–273. Mumbai (2007)

  19. Duflo, M., Serganova, V.: On Associated Variety for Lie Superalgebras. Preprint

  20. Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann. Math. 162(2), 581–642 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Freedman M., Kitaev A., Wang Z.: Simulation of topological field theories by quantum computers. Comm. Math. Phys. 227(3), 587–603 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Freedman M., Kitaev A., Larson M., Wang Z.: Topological quantum computation. Bull. Am. Math. Soc. (N.S.) 40(1), 31–38 (2003)

    Article  MATH  Google Scholar 

  23. Friedlander E., Parshall B.: Support varieties for restricted Lie algebras. Invent. Math. 86, 553–562 (1996)

    Article  MathSciNet  Google Scholar 

  24. Friedlander E., Parshall B.: Modular representation theory of Lie algebras. Amer. J. Math. 110(6), 1055–1093 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Friedlander E., Pevtsova J.: Π-supports for modules for finite group schemes. Duke Math. J. 139(2), 317–368 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ganter N., Kapranov M.: Representation and character theory in 2-categories. Adv. Math. 217(5), 2268–2300 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Geer, N., Patureau-Mirand, B.: Multivariable link invariants arising from Lie superalgebras of type I, preprint, arXiv:math/0609034v2, (2006)

  28. Geer N., Patureau-Mirand B.: An invariant trace for the category of representations of Lie superalgebras. Pacific J. Math. 238(2), 331–348 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Geer N., Patureau-Mirand B., Turaev V.: Modified quantum dimensions and re-normalized link invariants. Compos. Math. 145(1), 196–212 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Heller A., Reiner I.: Indecomposable representations. Ill. J. Math. 5, 314–323 (1961)

    MathSciNet  MATH  Google Scholar 

  31. Hovey, M., Palmieri, J., Strickland, N.: Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. 128(610) (1997)

  32. Kac V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)

    Article  MATH  Google Scholar 

  33. Kac, V.G.: Representations of classical Lie superalgebras, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), Lecture Notes in Math., vol. 676, pp. 597–626. Springer, Berlin (1978)

  34. Kac, V.G., Wakimoto, M.: Integrable highest weight modules over affine superalgebras and number theory, Lie theory and geometry. Progr. Math., vol. 123, pp. 415–456. Birkhäuser Boston, Boston, MA (1994)

  35. Kassel C.: Quantum Groups. Springer-Verlag GTM 155, Berlin (1995)

    MATH  Google Scholar 

  36. Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups I, preprint, arXiv:0803.4121v2, (2008)

  37. Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups II, preprint, arXiv:0804.2080, (2008)

  38. Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups III, preprint, arXiv:0807.3250, (2008)

  39. Knop F.: Tensor envelopes of regular categories. Adv. Math. 214(2), 571–617 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kremnizer, K.: Proof of the De Concini-Kac-Procesi conjecture, arXiv:math/0611236v1, (2006)

  41. Kujawa J.: Crystal structures arising from representations of GL(m|n). Represent. Theory 10, 49–85 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lauda, A.: A categorification of quantum sl(2), preprint, arXiv:0803.3652v3, (2008)

  43. Macdonald I.G.: Symmetric Functions and Hall Polynomials. 2nd edn. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  44. Moens E.M., Van Der Jeugt J.: A determinental formula for supersymmetric Schur polynomials. J. Alg. Comb. 17, 283–307 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. Okuyama, T.: A generalization of projective covers of modules over finite group algebras, unpublished manuscript.

  46. Premet, A.: The Green ring of a simple three-dimensional Lie p-algebra, (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. (10), 56–67; translation in Soviet Math. (Iz. VUZ) 35(10), 51–60 (1991)

  47. Premet A. Alexander: Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture. Invent. Math. 121(1), 79–117 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  48. Remmel, J.B.: The combinatorics of (k, l)-hook Schur functions. Combinatorics and algebra (Boulder, Colo., 1983), pp. 253–287, Contemp. Math., 34, Amer. Math. Soc., Providence, RI, (1984)

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

    Article  MathSciNet  MATH  Google Scholar 

  50. Rouquier, R.: 2-Kac-Moody algebras, preprint, arXiv:0812.5023v1, (2008)

  51. Serganova, V.: On Kac Wakimoto conjecture about dimension of simple representation of Lie superalgebra. Manaus, Brazil, July 7 (2009)

  52. Sergeev A.: Tensor algebra of the identity representation as a module over the Lie superalgebras GL(n, m) and Q(n). Math. USSR Sbornik 51, 419–427 (1985)

    Article  MATH  Google Scholar 

  53. Stembridge J.: Multiplicity-free products of Schur functions. Ann. Comb. 5(2), 113–121 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  54. Turaev V.G.: Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin (1994)

    Google Scholar 

  55. Wang, W., Zhao, L.: Representations of Lie superalgebras in prime characteristic I. Proc. London Math. Soc., to appear.

  56. Witten E.: Topological quantum field theory. Comm. Math. Phys. 117(3), 353–386 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  57. Witten E.: Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121(3), 351–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics & Statistics, Utah State University, Logan, UT, 84322, USA

    Nathan Geer

  2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Nathan Geer

  3. Mathematics Department, University of Oklahoma, Norman, OK, 73019, USA

    Jonathan Kujawa

  4. LMAM, université de Bretagne-Sud, université européenne de Bretagne, BP 573, 56017, Vannes, France

    Bertrand Patureau-Mirand

Authors
  1. Nathan Geer
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  2. Jonathan Kujawa
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  3. Bertrand Patureau-Mirand
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Correspondence to Nathan Geer.

Additional information

Research of the first author was partially supported by NSF grants DMS-0706725 and DMS-0968279.

Research of the second author was partially supported by NSF grant DMS-0734226.

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Geer, N., Kujawa, J. & Patureau-Mirand, B. Generalized trace and modified dimension functions on ribbon categories. Sel. Math. New Ser. 17, 453–504 (2011). https://doi.org/10.1007/s00029-010-0046-7

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