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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 mapfor the special linear groupSL_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

Sbe a punctured surface of finite genus. TheSL_2-skein algebra ofSis a non-commutative algebra whose elements are represented by framed linksKin the thickened surfaceS x [0,1]subject to certain relations. The skein algebra is a quantization of theSL_2(C)-character variety ofS, where the deformation depends on a complex parameterq. Bonahon and Wong constructed an injective algebra map, called the quantum trace, from the skein algebra ofSinto a simpler non-commutative algebra which can be thought of as a quantum Teichmüller space ofS. This map associates to a linkKinS x [0,1]a Laurentq-polynomial in non-commuting variablesX_i, which in the specializationq=1recovers the classical trace polynomial expressing the trace of monodromies of hyperbolic structures onSwhen 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 aSL_n-version of this invariant.