Topology Seminar (Main Talk): Genus Two Generalization of $A_1$ spherical Double Affine Hecke Algebra

Seminar | September 5 | 4-5 p.m. | 3 Evans Hall

 Semeon Artamonov, University of California, Berkeley

 Department of Mathematics

Spherical Double Affine Hecke Algebra can be viewed as a noncommutative \((q,t)\)-deformation of the \(SL(N,C)\) character variety of the fundamental group of a torus. This deformation inherits major topological property from its commutative counterpart, namely Mapping Class Group of a torus \(SL(2,Z)\) acts by atomorphisms of DAHA. In my talk I will define a genus two analogue of \(A_1\) spherical DAHA and show that the Mapping Class Group of a closed genus two surface acts by automorphisms of such algebra. I will then show that for special values of parameters \(q,t\) satisfying \(q^n t^2=1\) for some nonnegative integer n this algebra admits finite dimensional representations. I will conclude with discussion of potential applications to TQFT and knot theory.

Based on arXiv:1704.02947 joint with Sh. Shakirov