Dynamical quantum graphical calculus

Abstract: Graphical calculus provides a diagrammatic framework for performing topological computations with morphisms in strict tensor categories. The key idea is to identify such morphisms with oriented diagrams labeled by their in- and output objects. This was formalized by Reshetikhin and Turaev, by constructing for every strict tensor category $C$ a strict tensor functor that assigns isotopy classes of $C$-colored ribbon graphs to morphisms in $C$. This can be applied to the tensor category of finite-dimensional representations of a quantum group $U_q(g)$.

In this talk, I will first outline the fundamentals of this finite-dimensional quantum graphical calculus. Then I will explain how it can be extended to a larger category of quantum group representations, encompassing the quantum group analog of the BGG category $O$. In particular, this extended framework allows to visualize $U_q(g)$-intertwiners on Verma modules, as well as morphisms depending on a dynamical parameter, such as dynamical R-matrices. Finally, I will describe how this dynamical quantum graphical calculus can be used to obtain q-difference equations for quantum spherical functions.

This talk is based on joint work with Nicolai Reshetikhin (UC Berkeley) and Jasper Stokman (University of Amsterdam)

category theoryquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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