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.

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