Browse > Article
http://dx.doi.org/10.4134/JKMS.2012.49.5.993

THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS  

Jeong, Myeong-Ju (Korea Science Academy)
Kim, Dong-Seok (Department of Mathematics Kyonggi University)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.5, 2012 , pp. 993-1015 More about this Journal
Abstract
In this paper, we study the quantum sl($n$) representation category using the web space. Specially, we extend sl($n$) web space for $n{\geq}4$ as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial $P_n(q)$ specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl($n$). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial ($n=0$) and Jones polynomial ($n=2$) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.
Keywords
quantum sl($n$) representation theory; colored HOMFLY polynomial specialized to a one variable polynomial; periodic links; web spaces;
Citations & Related Records
연도 인용수 순위
  • Reference
1 I. Frenkel and M. Khovanov, Canonical bases in tensor products and graphical calculus for $U_{q}$(sl2), Duke Math. J. 87 (1997), no. 3, 409-480.   DOI
2 W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York-Heidelberg-Berlin, 1991.
3 V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1-25.   DOI
4 V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), no. 2, 335-388.   DOI
5 M.-J. Jeong and C.-Y. Park, Lens knots, periodic links and Vassiliev invariants, J. Knot Theory Ramifications 13 (2004), no. 8, 1041-1056.   DOI   ScienceOn
6 C. Kassel, M. Rosso, and V. Turaev, Quantum Groups and Knot Invariants, Panoramas et Syntheses, 5, Societe Mathematique de France, 1997.
7 M. Khovanov, sl(3) link homology, Algebr. Geom. Topol. 4 (2004), 1045-1081.   DOI
8 M. Khovanov, Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications 14 (2005), no. 1, 111-130.   DOI   ScienceOn
9 M. Khovanov, private communication.
10 M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008), no. 1, 1-91.   DOI
11 D. Kim, Graphical Calculus on Representations of Quantum Lie Algebras, Thesis, UC-Davis, 2003, arXiv:math.QA/0310143.
12 D. Kim and J. Lee, The quantum sl(3) invariants of cubic bipartite planar graphs, J. Knot Theory Ramifications 17 (2008), no. 3, 361-375.   DOI   ScienceOn
13 C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, New York, W. H. Freeman, 1994.
14 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.   DOI   ScienceOn
15 S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves I. The SL(2) case, Duke Math. J. 142 (2008), no. 3, 511-588.   DOI
16 N. Chbili, The quantum SU(3) invariant of links and Murasugi's congruence, Topology Appl. 122 (2002), no. 3, 479-485.
17 J. Przytycki and A. Sikora, On skein algebras and $Sl_{2}$(C)-character varieties, Topology 39 (2000), no. 1, 115-148.   DOI   ScienceOn
18 N. Chbili, Quantum invariants and finite group actions on three-manifolds, Topology Appl. 136 (2004), no. 1-3, 219-231.   DOI   ScienceOn
19 Y. Yokota, The Kauffman polynomial of periodic knots, Topology 32 (1993), no. 2, 309-324.   DOI   ScienceOn
20 Y. Yokota, Skein and quantum SU(N) invariants of 3-manifolds, Math. Ann. 307 (1997), no. 1, 109-138.   DOI
21 N. Yu. Reshetikhin and V. G. Turaev, Ribbob graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1-26.   DOI
22 N. Yu. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547-597.   DOI
23 T. Van Zandt, PSTricks: PostScript macros for generic $T_{E}X$, Available at ftp://ftp.princeton.edu/pub/tvz/.
24 A. Sikora and B. Westbury, Confluence theory for graphs, Algebr. Geom. Topol. 7 (2007), 439-478.   DOI
25 P. Traczyk, A criterion for knots of period 3, Topology Appl. 36 (1990), no. 3, 275-281.   DOI   ScienceOn
26 V. G. Turaev, The Conway and Kauffman modules of a solid torusa, (translation) J. Soviet Math. 52 (1990), no. 1, 2799-2805.   DOI
27 M. Vybornov, Solutions of the Yang-Baxter equation and quantum sl(2), J. Knot Theory Ramifications 8 (1999), no. 7, 953-961.   DOI
28 H. Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no. 1, 5-9.
29 B. Westbury, Invariant tensors for the spin representation of so(7), Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 1, 217-240.
30 E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351-399.   DOI
31 Y. Yokota, The skein polynomial of periodic knots, Math. Ann. 291 (1991), no. 2, 281-291.   DOI
32 Y. Yokota, The Jones polynomial of periodic knots, Proc. Amer. Math. Soc. 113 (1991), no. 3, 889-894.   DOI   ScienceOn
33 R. Kirby and P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2), Invent. Math. 105 (1991), no. 3, 473-545.   DOI
34 G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), no. 1, 109-151.   DOI
35 K. Murasugi, On periodic knots, Comment. Math. Helv. 46 (1971), 162-174.   DOI
36 T. Le, Integrality and symmetry of quantum link invariants, Duke Math. J. 102 (2000), no. 2, 273-306.   DOI
37 W. Lickorish, Distinct 3-manifolds with all SU(2)q invariants the same, Proc. Amer. Math. Soc. 117 (1993), no. 1, 285-292.
38 S. Morrison, A Diagrammatic Category for the Representation Theory of $U_{q}$($sl_{n}$), UC Berkeley Ph.D. thesis, arXiv:0704.1503.
39 K. Murasugi, Jones polynomials of periodic links, Pacific J. Math. 131 (1988), no. 2, 319-329.   DOI
40 H. Murakami, Asymptotic Behaviors of the colored Jones polynomials of a torus knot, Internat. J. Math. 15 (2004), no. 6, 547-555.   DOI   ScienceOn
41 H.Murakami, T. Ohtsuki, and S. Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. (2) 44 (1998), no. 3-4, 325-360.
42 T. Ohtsuki and S. Yamada, Quantum SU(3) invariant of 3-manifolds via linear skein theory, J. Knot Theory Ramifications 6 (1997), no. 3, 373-404.   DOI   ScienceOn
43 J. H. Przytycki, On Murasugi's and Traczyk's criteria for periodic links, Math. Ann. 283 (1989), no. 3, 465-478.   DOI
44 J. Przytycki and A. Sikora, $SU_{n}$-quantum invariants for periodic links, Diagrammatic morphisms and applications (San Francisco, CA, 2000), 199-205, Contemp. Math., 318, Amer. Math. Soc., Providence, RI, 2003.
45 Q. Chen and T. Le, Quantum invariants of periodic links and periodic 3-manifolds, Fund. Math. 184 (2004), 55-71.   DOI