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Classical and quantum traces coming from SL_n(C) and U_q(sl_n)

QMAP Seminar

Speaker: Daniel Douglas, University of Southern California
Related Webpage: https://dornsife.usc.edu/mathematics/graduateassistants/
Location: 2112 MSB
Start time: Fri, Nov 22 2019, 11:00AM

We discuss work-in-progress constructing a quantum trace map for the special linear group SL_n. This is a kind of Reshetikhin-Turaev invariant for knots in thickened punctured surfaces, coming from an interaction between higher Teichmüller theory and quantum groups.

Let S be a punctured surface of finite genus. The SL_2-skein algebra of S is a non-commutative algebra whose elements are represented by framed links K in the thickened surface S x [0,1] subject to certain relations. The skein algebra is a quantization of the SL_2(C)-character variety of S, where the deformation depends on a complex parameter q. Bonahon and Wong constructed an injective algebra map, called the quantum trace, from the skein algebra of S into a simpler non-commutative algebra which can be thought of as a quantum Teichmüller space of S. This map associates to a link K in S x [0,1] a Laurent q-polynomial in non-commuting variables X_i, which in the specialization q=1 recovers the classical trace polynomial expressing the trace of monodromies of hyperbolic structures on S when written in Thurston's shear-bend coordinates for Teichmüller space. In the early 2000s, Fock and Goncharov, among others, developed a higher Teichmüller theory, which should lead to a SL_n-version of this invariant.