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The Witten-Reshetikhin-Turaev representation of the Kauffman bracket skein algebra


Authors: Francis Bonahon and Helen Wong
Journal: Proc. Amer. Math. Soc. 144 (2016), 2711-2724
MSC (2010): Primary 57M27, 57R56
DOI: https://doi.org/10.1090/proc/12927
Published electronically: November 30, 2015
MathSciNet review: 3477089
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Abstract: For $ A$ a primitive $ 2N$-root of unity with $ N$ odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman bracket skein algebra of a closed surface. We show that this representation is irreducible, and we compute its classical shadow in the sense of an earlier work of the authors.


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  • [1] C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992), no. 4, 685-699. MR 1191373 (94a:57010), https://doi.org/10.1016/0040-9383(92)90002-Y
  • [2] C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), no. 4, 883-927. MR 1362791 (96i:57015), https://doi.org/10.1016/0040-9383(94)00051-4
  • [3] F. Bonahon and H. Wong, Kauffman brackets, character varieties and triangulations of surfaces, Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 179-194. MR 2866931, https://doi.org/10.1090/conm/560/11099
  • [4] F. Bonahon and H. Wong, Representations of the Kauffman bracket skein algebra I: invariants and punctured surfaces, to appear in Inventiones Mathematicæ, arXiv:1206.1638, 2012.
  • [5] F. Bonahon and H. Wong, Representations of the Kauffman bracket skein algebra II: punctured surfaces, preprint, arXiv:1206.1639, 2012.
  • [6] F. Bonahon and H. Wong, Representations of the Kauffman bracket skein algebra III: closed surfaces and canonicity, preprint, arXiv:1505.01522, 2015.
  • [7] D. Bullock, C. Frohman, and J. Kania-Bartoszyńska, Understanding the Kauffman bracket skein module, J. Knot Theory Ramifications 8 (1999), no. 3, 265-277. MR 1691437 (2000d:57012), https://doi.org/10.1142/S0218216599000183
  • [8] D. Bullock, C. Frohman, and J. Kania-Bartoszyńska, The Kauffman bracket skein as an algebra of observables, Proc. Amer. Math. Soc. 130 (2002), no. 8, 2479-2485 (electronic). MR 1897475 (2003e:57016), https://doi.org/10.1090/S0002-9939-02-06323-2
  • [9] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335-388. MR 908150 (89c:46092), https://doi.org/10.2307/1971403
  • [10] L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395-407. MR 899057 (88f:57006), https://doi.org/10.1016/0040-9383(87)90009-7
  • [11] L. H. Kauffman and S. L. Lins, Temperley-Lieb recoupling theory and invariants of $ 3$-manifolds, Annals of Mathematics Studies, vol. 134, Princeton University Press, Princeton, NJ, 1994. MR 1280463 (95c:57027)
  • [12] T. T. Q. Lê, On Kauffman bracket skein modules at roots of unity, Algebr. Geom. Topol. 15 (2015), no. 2, 1093-1117. MR 3342686, https://doi.org/10.2140/agt.2015.15.1093
  • [13] W. B. R. Lickorish, The skein method for three-manifold invariants, J. Knot Theory Ramifications 2 (1993), no. 2, 171-194. MR 1227009 (94g:57006), https://doi.org/10.1142/S0218216593000118
  • [14] W. B. R. Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. MR 1472978 (98f:57015)
  • [15] W. B. R. Lickorish, Quantum invariants of 3-manifolds, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 707-734. MR 1886680 (2003d:57026)
  • [16] G. Masbaum and P. Vogel, $ 3$-valent graphs and the Kauffman bracket, Pacific J. Math. 164 (1994), no. 2, 361-381. MR 1272656 (95e:57003)
  • [17] J. H. Przytycki, Skein modules of $ 3$-manifolds, Bull. Polish Acad. Sci. Math. 39 (1991), no. 1-2, 91-100. MR 1194712 (94g:57011)
  • [18] J. H. Przytycki and A. S. Sikora, On skein algebras and $ {\rm Sl}_2({\bf C})$-character varieties, Topology 39 (2000), no. 1, 115-148. MR 1710996 (2000g:57026), https://doi.org/10.1016/S0040-9383(98)00062-7
  • [19] N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1-26. MR 1036112 (91c:57016)
  • [20] N. Reshetikhin and V. G. Turaev, Invariants of $ 3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547-597. MR 1091619 (92b:57024), https://doi.org/10.1007/BF01239527
  • [21] J. Roberts, Skeins and mapping class groups, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 53-77. MR 1253282 (94m:57035), https://doi.org/10.1017/S0305004100071917
  • [22] J. Roberts, Irreducibility of some quantum representations of mapping class groups, J. Knot Theory Ramifications 10 (2001), no. 5, 763-767. Knots in Hellas '98, Vol. 3 (Delphi). MR 1839700 (2002f:57065), https://doi.org/10.1142/S021821650100113X
  • [23] V. G. Turaev, The Conway and Kauffman modules of a solid torus, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), no. Issled. Topol. 6, 79-89, 190 (Russian, with English summary); English transl., J. Soviet Math. 52 (1990), no. 1, 2799-2805. MR 964255 (90f:57012), https://doi.org/10.1007/BF01099241
  • [24] V. G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 6, 635-704. MR 1142906 (94a:57023)
  • [25] V. G. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673 (95k:57014)
  • [26] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351-399. MR 990772 (90h:57009)

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Additional Information

Francis Bonahon
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: fbonahon@math.usc.edu

Helen Wong
Affiliation: Department of Mathematics, Carleton College, Northfield, Minnesota 55057
Email: hwong@carleton.edu

DOI: https://doi.org/10.1090/proc/12927
Received by editor(s): January 1, 2011
Received by editor(s) in revised form: July 17, 2015, and January 1, 2015
Published electronically: November 30, 2015
Additional Notes: This research was partially supported by grants DMS-0604866, DMS-1105402 and DMS-1105692 from the National Science Foundation, and by a mentoring grant from the Association for Women in Mathematics.
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2015 American Mathematical Society

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