A graphical approach to the Drinfeld centre

Abstract

Let $\mathcal{C}$ be a spherical fusion category. The goal of this article is to present the tube category of $\mathcal{C}$, denoted $\mathcal{T}\mathcal{C}$, as giving an alternative graphical perspective on the Drinfeld centre of $\mathcal{C}$, denoted $Z(\mathcal{C})$. We then exploit this perspective to obtain an alternative proof of the equivalence between $Z(\mathcal{C})$ and $\mathcal{C}⊠\bar{\mathcal{C}}$ when $\mathcal{C}$ is modular. Sections 2-4 provide a survey of the required prerequisites; readers already familiar with graphical calculus in fusion categories should start at Section 5.

29 pages; v2: restructured the survey portion of the paper (sections 2-4) and added certain references