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

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.Contents 1.5. The ribbon element and twisting in category M 1.6. Dual representations 2. Graphical calculus 2.1. Ribbon graphs and ribbon graph diagrams 2.2. Ribbon-braid graphs and ribbon-braid graph diagrams 2.3. Strictifications 2.4. Graphical calculus for D-colored ribbon-braid graphs 2.5. The Reshetikhin-Turaev functor 2.6. Colored ribbon graph subdiagrams in B D 2.7. Fusion morphisms 2.8. Bundling of strands 3. q-KZ equations for k-point dynamical fusion operators 3.1. k-point quantum vertex operators and their graphical representation 3.2. The topological operator q-KZ equations 3.3. The topological q-KZ equations for the k-point dynamical fusion operator Conclusion References