Title: Quantum symmetries old and new.
 

Abstract: Quantum symmetries is the study of how symmetries of classical objects carry over to their quantizations. 

In this lecture I will review the work of De Concini, Kac and Lusztig on how the coadjoint action of an algebra group G carries over to an action of a quantum group at a root of unity on its restricted dual at that root of unity.

I will then resolve a recent conjecture in skein theory negatively by analyzing the quantum symmetries of the stated skein module and offer a corrected version of the conjecture. 
More explicitly:
The stated skein module of a compact oriented three-manifold at a root of unity is the global sections of a sheaf over the SL_2C representation variety of the fundamental group of the manifold.  Patrick Kinnear conjectured that it is an equivariant line bundle.

I will explicate how the action of SL_2C by conjugation on the representation variety  carries over to symmetries of the stated skein module. The quantum symmetries turn  out to be modulated by Lusztig’s restricted dual specialized at the root of unity. From there I can construct examples where the sheaf underlying the stated skein module is not SL_2C equivariant, and identify a canonical subsheaf that is SL_2C-equivariant.