Title: The Family of Induced Representations for Quantum Groups at a 4th Root of Unity
Speaker: Matthew Harper (OSU)
Abstract: For any non-zero complex number, we can construct an induced representation of $U_\xi(\mathfrak{sl}_2)$ for $\xi$ a root of unity. As shown by Ohtsuki, these representations can be used to define a knot invariant, in this case it is Alexander polynomial. We will go through the steps of this construction and examine how the representation theory imposes the skein relation. From the rank one case, we build the analogous rank two invariant, and classify the reducibility of these representations. Finally, we examine the tensor product structure of these representations, the resulting "$\mathfrak{sl}_3$" skein relation, and an approach on how to generalize further to higher rank quantum groups.
Seminar URL: https://research.math.osu.edu/reps/
