Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 030, 36 pages      arXiv:1309.4985      https://doi.org/10.3842/SIGMA.2015.030
Contribution to the Special Issue on New Directions in Lie Theory

Skein Modules from Skew Howe Duality and Affine Extensions

Hoel Queffelec
Mathematical Sciences Institute, The Australian National University, J.D. 27 Union Lane, Acton ACT 2601, Australia

Received July 22, 2014, in final form March 30, 2015; Published online April 15, 2015

Abstract
We show that we can release the rigidity of the skew Howe duality process for ${\mathfrak sl}_n$ knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine ${\mathfrak sl}_m$ case, corresponding to looking at tangles embedded in a solid torus. We investigate the relations between the invariants constructed by evaluation representations (and affinization of them) and usual skein modules, and give tools for interpretations of annular skein modules as sub-algebras of intertwiners for particular $U_q({\mathfrak sl}_n)$ representations. The categorification proposed in a joint work with A. Lauda and D. Rose also admits a direct extension in the affine case.

Key words: skein modules; quantum groups; annulus; affine quantum groups.

pdf (640 kb)   tex (51 kb)

References

  1. Bar-Natan D., Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443-1499, math.GT/0410495.
  2. Blanchet C., An oriented model for Khovanov homology, J. Knot Theory Ramifications 19 (2010), 291-312, arXiv:1405.7246.
  3. Carter J.S., Reidemeister/Roseman-type moves to embedded foams in 4-dimensional space, arXiv:1210.3608.
  4. Cautis S., Clasp technology to knot homology via the affine Grassmannian, arXiv:1207.2074.
  5. Cautis S., Kamnitzer J., Licata A., Categorical geometric skew Howe duality, Invent. Math. 180 (2010), 111-159, arXiv:0902.1795.
  6. Cautis S., Kamnitzer J., Morrison S., Webs and quantum skew Howe duality, arXiv:1210.6437.
  7. Cautis S., Lauda A.D., Implicit structure in 2-representations of quantum groups, Selecta Math. 21 (2015), 201-244, arXiv:1111.1431.
  8. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1995.
  9. Clark D., Morrison S., Walker K., Fixing the functoriality of Khovanov homology, Geom. Topol. 13 (2009), 1499-1582, math.GT/0701339.
  10. Cooper B., Krushkal V., Categorification of the Jones-Wenzl projectors, Quantum Topol. 3 (2012), 139-180, arXiv:1005.5117.
  11. Doty S.R., Green R.M., Presenting affine $q$-Schur algebras, Math. Z. 256 (2007), 311-345, math.QA/0603002.
  12. Frenkel I., Stroppel C., Sussan J., Categorifying fractional Euler characteristics, Jones-Wenzl projectors and $3j$-symbols, Quantum Topol. 3 (2012), 181-253, arXiv:1007.4680.
  13. Grant J., The moduli problem of Lobb and Zentner and the colored $\mathfrak{sl}(N)$ graph invariant, J. Knot Theory Ramifications 22 (2013), 1350060, 16 pages, arXiv:1212.4511.
  14. Green R.M., The affine $q$-Schur algebra, J. Algebra 215 (1999), 379-411, q-alg/9705015.
  15. Hong J., Kang S.J., Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, Vol. 42, Amer. Math. Soc., Providence, RI, 2002.
  16. Kamnitzer J., Tingley P., The crystal commutor and Drinfeld's unitarized $R$-matrix, J. Algebraic Combin. 29 (2009), 315-335, arXiv:0707.2248.
  17. Kauffman L.H., Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989), 697-710.
  18. Khovanov M., A categorification of the Jones polynomial, Duke Math. J. 101 (2000), 359-426, math.QA/9908171.
  19. Khovanov M., A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665-741, math.QA/0103190.
  20. Khovanov M., ${\rm sl}(3)$ link homology, Algebr. Geom. Topol. 4 (2004), 1045-1081, math.QA/0304375.
  21. Khovanov M., Lauda A.D., A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347, arXiv:0803.4121.
  22. Khovanov M., Lauda A.D., A categorification of quantum ${\rm sl}(n)$, Quantum Topol. 1 (2010), 1-92, arXiv:0807.3250.
  23. Khovanov M., Lauda A.D., A diagrammatic approach to categorification of quantum groups. II, Trans. Amer. Math. Soc. 363 (2011), 2685-2700, arXiv:0804.2080.
  24. Khovanov M., Lauda A.D., Mackaay M., Stošić M., Extended graphical calculus for categorified quantum ${\rm sl}(2)$, Mem. Amer. Math. Soc. 219 (2012), vi+87 pages, arXiv:1006.2886.
  25. Kim D., Graphical calculus on representations of quantum Lie algebras, Ph.D. Thesis, University of California, Davis, 2003.
  26. Kuperberg G., Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180 (1996), 109-151, q-alg/9712003.
  27. Lauda A.D., A categorification of quantum ${\rm sl}(2)$, Adv. Math. 225 (2010), 3327-3424, arXiv:0803.3652.
  28. Lauda A.D., Queffelec H., Rose D.E.V., Khovanov homology is a skew Howe 2-representation of categorified quantum $\mathfrak{sl}_m$, Algebr. Geom. Topol., to appear, arXiv:1212.6076.
  29. Lusztig G., Introduction to quantum groups, Progress in Mathematics, Vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993.
  30. Mackaay M., Stošić M., Vaz P., The $1,2$-coloured HOMFLY-PT link homology, Trans. Amer. Math. Soc. 363 (2011), 2091-2124, arXiv:0809.0193.
  31. Mackaay M., Thiel A.L., Categorifications of the extended affine Hecke algebra and the affine $q$-Schur algebra $S(n,r)$ for $2arXiv:1302.3102.
  32. Mazorchuk V., Stroppel C., A combinatorial approach to functorial quantum $\mathfrak{sl}_k$ knot invariants, Amer. J. Math. 131 (2009), 1679-1713, arXiv:0709.1971.
  33. Morrison S.E., A diagrammatic category for the representation theory of $U_q(\mathfrak{sl}_n)$, Ph.D. Thesis, University of California, Berkeley, 2007.
  34. Murakami H., Ohtsuki T., Yamada S., Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. 44 (1998), 325-360.
  35. Rose D.E.V., A categorification of quantum $\mathfrak{sl}_3$ projectors and the $\mathfrak{sl}_3$ Reshetikhin-Turaev invariant of tangles, Quantum Topol. 5 (2014), 1-59, arXiv:1109.1745.
  36. Stern E., Semi-infinite wedges and vertex operators, Int. Math. Res. Not. 1995 (1995), no. 4, 201-219, q-alg/9504013.
  37. Stroppel C., Sussan J., Categorified Jones-Wenzl projectors: a comparison, arXiv:1105.3038.
  38. Takemura K., Uglov D., Level-$0$ action of $U_q(\widehat{\mathfrak{sl}}_n)$ on the $q$-deformed Fock spaces, Comm. Math. Phys. 190 (1998), 549-583, q-alg/9607031.
  39. Uglov D., Canonical bases of higher-level $q$-deformed Fock spaces and Kazhdan-Lusztig polynomials, in Physical Combinatorics (Kyoto, 1999), Progr. Math., Vol. 191, Birkhäuser Boston, Boston, MA, 2000, 249-299, math.QA/9905196.

Previous article  Next article   Contents of Volume 11 (2015)