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Ambidextrous objects and trace functions for nonsemisimple categories


Authors: Nathan Geer, Jonathan Kujawa and Bertrand Patureau-Mirand
Journal: Proc. Amer. Math. Soc. 141 (2013), 2963-2978
MSC (2010): Primary 18D10; Secondary 17B99, 16T05, 20C20, 57M99
DOI: https://doi.org/10.1090/S0002-9939-2013-11563-7
Published electronically: May 10, 2013
MathSciNet review: 3068949
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Abstract: We provide a necessary and sufficient condition for a simple object in a pivotal $ \Bbbk $-category to be ambidextrous. In turn, these objects imply the existence of nontrivial trace functions in the category. These functions play an important role in low-dimensional topology as well as in studying the category itself. In particular, we prove they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the $ (1,p)$ minimal model in conformal field theory, and quantum groups at a root of unity.


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  • 1. J. L. Alperin, Local representation theory: Modular representations as an introduction to the local representation theory of finite groups, Cambridge Studies in Advanced Mathematics, 11, Cambridge University Press, Cambridge, 1986. MR 860771 (87i:20002)
  • 2. H. H. Andersen, P. Polo, K. Wen, Representations of quantum algebras, Invent. Math. 104 (1991), no. 1, 1-59. MR 1094046 (92e:17011)
  • 3. B. Bakalov, A. Kirillov, Jr. Lectures on tensor categories and modular functors, University Lecture Series, 21, American Mathematical Society, Providence, RI, 2001. MR 1797619 (2002d:18003)
  • 4. J. Barrett, B. Westbury, Spherical categories, Adv. Math. 143 (1999), 357-375. MR 1686423 (2000c:18007)
  • 5. C. Bendel, Generalized reduced enveloping algebras for restricted Lie algebras, J. Algebra 218 (1999), no. 2, 373-411. MR 1705814 (2000h:17007)
  • 6. D. J. Benson, 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. MR 1644252 (99f:20001a)
  • 7. J. Bichon, Cosovereign Hopf algebras, J. Pure Appl. Algebra 157 (2001), no. 2-3, 121-133. MR 1812048 (2001m:16058)
  • 8. B. Boe, J. Kujawa, D. Nakano, Complexity for modules over the classical Lie superalgebra $ \mathfrak{gl}(m\vert n)$, Compositio Math. 148 (2012), no. 5, 1561-1592. MR 2982440
  • 9. B. Boe, J. Kujawa, D. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. (2011), issue 3, 696-724. MR 2764876
  • 10. J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $ \mathfrak{q}(n)$, Adv. in Math. 182 (2004), 28-77. MR 2028496 (2004m:17018)
  • 11. V. Chari, A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632 (95j:17010)
  • 12. M. Cohen, S. Westreich, Characters and a Verlinde-type formula for symmetric Hopf algebras, J. Algebra 320 (2008), no. 12, 4300-4316. MR 2464107 (2010b:16070)
  • 13. J. Comes, J. Kujawa, Modified Traces on Deligne's Category $ \operatorname {Rep}(S_{t})$, J. Algebraic Combin. 36 (2012), no. 4, 541-560. MR 2984156
  • 14. J. Comes, V. Ostrik, On blocks of Deligne's category $ \underline {\operatorname {Re}}\operatorname {p}(S_t)$, Adv. in Math. 226 (2011), no. 2, 1331-1377. MR 2737787
  • 15. P. Etingof, D. Nikshych, V. Ostrik, An analogue of Radford's $ S^{4}$ formula for finite tensor categories, Int. Math. Res. Not. (2004), no. 54, 2915-2933. MR 2097289 (2005m:18007)
  • 16. P. Etingof, V. Ostrik, Finite tensor categories, Mosc. Math. J. 4 (2004), no. 3, 627-654. MR 2119143 (2005j:18006)
  • 17. R. Farnsteiner, On Frobenius extensions defined by Hopf algebras, J. Algebra 166 (1994), no. 1, 130-141. MR 1276820 (95e:16034)
  • 18. J. Fuchs, On non-semisimple fusion rules and tensor categories, Lie algebras, vertex operator algebras and their applications, 315-337, Contemp. Math., 442, Amer. Math. Soc., Providence, RI, 2007. MR 2372571 (2009h:81261)
  • 19. N. Geer, R. Kashaev, V. Turaev, Tetrahedral forms in monoidal categories and 3-manifold invariants, J. Reine Angew. Math. 673 (2012), 69-123. MR 2999129
  • 20. N. Geer, J. Kujawa, B. Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories, Selecta Math. 17 (2011), no. 2, 453-504. MR 2803849
  • 21. N. Geer, B. Patureau-Mirand, Topological invariants from non-restricted quantum groups, arXiv:1009.4120 (2010).
  • 22. -, Multivariable link invariants arising from Lie superalgebras of type I, J. of Knot Theory and its Ramifications 19 (2010), no. 1, 93-115. MR 2640994 (2011h:57012)
  • 23. N. Geer, B. Patureau-Mirand, V. Turaev, Modified $ 6j$-symbols and $ 3$-manifold invariants, Adv. Math. 228 (2011), no. 2, 1163-1202. MR 2822220 (2012m:57019)
  • 24. N. Geer, B. Patureau-Mirand, A. Virelizier, Traces on ideals in pivotal categories, Quantum Topology 4 (2013), 91-124. MR 2998839
  • 25. M. Gorelik, On the ghost centre of Lie superalgebras, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 6, 1745-1764. MR 1817382 (2002c:17017)
  • 26. A. Granville, K. Ono, Defect zero $ p$-blocks for finite simple groups, Trans. Amer. Math. Soc. 348 (1996), no. 1, 331-347. MR 1321575 (96e:20014)
  • 27. Y.-Z. Huang, J. Lepowsky, and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, arXiv:1012.4193 (2010).
  • 28. J. E. Humphreys, Symmetry for finite dimensional Hopf algebras, Proc. Amer. Math. Soc. 68 (1978), no. 2, 143-146. MR 0485965 (58:5757)
  • 29. J. Jantzen, Representations of algebraic groups, second edition, Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003. MR 2015057 (2004h:20061)
  • 30. L. Kauffman, D. Radford, A necessary and sufficient condition for a finite dimensional Drinfeld double to be a ribbon Hopf algebra, J. Algebra 159 (1993), no. 1, 98-114. MR 1231205 (94d:16037)
  • 31. V. G. Kac, Lie superalgebras, Adv. in Math. 26 (1977), no. 1, 8-96. MR 0486011 (58:5803)
  • 32. -, Characters of typical representations of classical Lie superalgebras, Comm. Algebra 5 (1977), no. 8, 889-897. MR 0444725 (56:3075)
  • 33. V. Kac, M. Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, Lie theory and geometry, Progr. Math., 123, Birkhäuser Boston, Boston, MA, 1994, 415-456. MR 1327543 (96j:11056)
  • 34. C. Kassel, Quantum groups, Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995. MR 1321145 (96e:17041)
  • 35. S. Kumar, Representations of quantum groups at roots of unity, Proc. of the Conf. on Quantum Topology (Manhattan, KS, 1993), 187-224, World Sci. Publ., River Edge, NJ, 1994. MR 1309933 (95m:17006)
  • 36. R. G. Larson, M. E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75-94. MR 0240169 (39:1523)
  • 37. M. Lorenz, Representations of finite-dimensional Hopf algebras, J. Algebra 188 (1997), no. 2, 476-505. MR 1435369 (98i:16039)
  • 38. V. Mazorchuk, V. Miemietz, Serre functors for Lie algebras and superalgebras, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 1, 47-75. MR 2986264
  • 39. S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 82, American Mathematical Society, Providence, RI, 1993. MR 1243637 (94i:16019)
  • 40. K. Nagatomo, A. Tsuchiya, The triplet vertex operator algebra $ W(p)$ and the restricted quantum group $ \overline {U}_q(\mathfrak{sl}_2)$ at $ q=e^{\tfrac {\pi i}{p}}.$ Exploring new structures and natural constructions in mathematical physics, Adv. Stud. Pure Math., 61, Math. Soc. Japan, Tokyo, 2011, 1-49. MR 2867143
  • 41. D. Radford, Minimal quasitriangular Hopf algebras, J. Algebra 157 (1993), no. 2, 285-315. MR 1220770 (94c:16052)
  • 42. U. Oberst, H-J. Schneider, Über Untergruppen endlicher algebraischer Gruppen, Manuscripta Math. 8 (1973), 217-241. MR 0347838 (50:339)
  • 43. V. Serganova, On the superdimension of an irreducible representation of a basic classical Lie superalgebra, Supersymmetry in Mathematics and Physics, Springer LNM, 2027, eds., S. Ferrara, R. Fioresi, V.S. Varadarajan, Springer, Heidelberg, 2011. MR 2906346 (2012m:17014)
  • 44. I. Tsohantjis, M. D. Gould, Quantum double finite group algebras and link polynomials, Bull. Austral. Math. Soc. 49 (1994), no. 2, 177-204. MR 1265357 (95a:57016)
  • 45. V.G. Turaev, Quantum invariants of knots and $ 3$-manifolds, de Gruyter Studies in Mathematics, 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673 (95k:57014)
  • 46. W. Wang, L. Zhao, Representations of Lie superalgebras in prime characteristic. II: The queer series, J. Pure Appl. Algebra 215 (2011), no. 10, 2515-2532. MR 2793955
  • 47. -, Representations of Lie superalgebras in prime characteristic. I, Proc. Lond. Math. Soc. (3) 99 (2009), no. 1, 145-167. MR 2520353 (2011a:17032)
  • 48. S. Witherspoon, The representation ring of the quantum double of a finite group, J. Algebra 179 (1996), 305-329. MR 1367852 (96m:20015)
  • 49. L. Zhao, Representations of Lie superalgebras in prime characteristic. III, Pacific J. Math. 248 (2010), no. 2, 493-510. MR 2741259

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Additional Information

Nathan Geer
Affiliation: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322
Email: nathan.geer@usu.edu

Jonathan Kujawa
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: kujawa@math.ou.edu

Bertrand Patureau-Mirand
Affiliation: LMAM, Université de Bretagne-Sud, Université Européenne de Bretagne, BP 573, 56017 Vannes, France
Email: bertrand.patureau@univ-ubs.fr

DOI: https://doi.org/10.1090/S0002-9939-2013-11563-7
Received by editor(s): June 22, 2011
Received by editor(s) in revised form: November 15, 2011
Published electronically: May 10, 2013
Additional Notes: Research of the first author was partially supported by NSF grants DMS-0968279 and DMS-1007197.
Research of the second author was partially supported by NSF grant DMS-0734226 and NSA grant H98230-11-1-0127.
Communicated by: Kailash E. Misra
Article copyright: © Copyright 2013 American Mathematical Society

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