Graphical calculus for quantum vertex operators, I: The dynamical fusion operator

TL;DR

This paper develops a graphical calculus framework for quantum vertex operators within ribbon and braided monoidal categories, extending the linear operator equations for dynamical fusion operators to a q-KZ system, with applications to quantum group modules.

Contribution

It introduces a detailed graphical calculus approach for quantum vertex operators and extends existing operator equations to a q-KZ type system for quantum group modules.

Findings

Established foundational graphical calculus for ribbon and braided categories.

Extended dynamical fusion operator equations to a q-KZ system.

Applied the framework to categories of quantum group modules.

Abstract

This paper is the first in a series on graphical calculus for quantum vertex operators. We establish in great detail the foundations of graphical calculus for ribbon categories and braided monoidal categories with twist. We illustrate the potential of this approach by applying it to various categories of quantum group modules, in particular to derive an extension of the linear operator equation for dynamical fusion operators, due to Arnaudon, Buffenoir, Ragoucy and Roche, to a system of linear operator equations of $q$-KZ type.