[Paper Review] A graphical approach to the Drinfeld centre

Leonard Hardiman|arXiv (Cornell University)|Nov 17, 2019

Homotopy and Cohomology in Algebraic Topology7 references4 citations

TL;DR

This paper presents a graphical framework for the Drinfeld centre Z(C) of a spherical fusion category C using the tube category TC, which provides a topological interpretation of half-braiding structures via idempotents on a cylinder. It offers a new proof of the equivalence between Z(C) and C ⊠ C when C is modular, using graphical calculus to show that the image of the canonical functor Φ: C ⊠ C → Z(C) corresponds to a complete set of orthogonal primitive idempotents in TC, thereby establishing equivalence via idempotent completion and dimension counting.

ABSTRACT

Let $\mathcal{C}$ be a spherical fusion category. The goal of this article is to present the tube category of $\mathcal{C}$, denoted $\mathcal{TC}$, 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 $\mathcal{C} \boxtimes \bar{\mathcal{C}}$ 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.

Motivation & Objective

• To provide a graphical, topological interpretation of the Drinfeld centre Z(C) of a spherical fusion category C using the tube category TC.
• To establish a correspondence between objects in Z(C) and idempotents in TC via the Yoneda embedding and half-braiding data.
• To offer a new, diagrammatic proof of the equivalence between Z(C) and C ⊠ C when C is modular, using the tube category's structure.
• To formalize the connection between Ocneanu's tube algebra and the categorical structure of Z(C) through idempotent completion.

Proposed method

• Define the tube category TC as a category whose objects are those of C and whose morphisms are diagrams on a cylinder, encoding half-braiding data.
• Construct an idempotent ǫτ ∈ EndTC(X) for each half-braiding τ on X, which classifies the image of (X, τ) ∈ Z(C) under the Yoneda embedding.
• Use the Yoneda embedding to show that RT C (the idempotent completion of TC) is equivalent to Z(C), via the correspondence between half-braiding and idempotent structure.
• Apply graphical calculus to the idempotent ǫYX = 1/d(C) ⊗S d(S) · (X ⊗Y →S →X ⊗Y) to represent the image of X ⊠ Y under the functor Φ: C ⊠ C → Z(C).
• Prove that the set {ǫJ_I}I,J∈Irr(C) forms a complete set of orthogonal primitive idempotents in TC by computing HomT C(ǫYX, ǫBA) and showing it is isomorphic to HomC(X,A) ⊗ HomC(Y,B).
• Use dimension counting and the killing ring lemma to verify that the composition map ∑IJ HomT C(X, ǫJ_I) ⊗ HomT C(ǫJ_I, Y) → HomT C(X,Y) is an isomorphism, establishing fullness and essential surjectivity of Φ.

Experimental results

Research questions

• RQ1 How can the Drinfeld centre Z(C) of a spherical fusion category C be interpreted graphically using topological diagrams on a cylinder?
• RQ2 What is the categorical role of the tube category TC in realizing the Drinfeld centre via idempotent completion?
• RQ3 How does the graphical idempotent ǫτ ∈ EndTC(X) correspond to a half-braiding on X in Z(C)?
• RQ4 Can the equivalence Z(C) ≅ C ⊠ C for modular categories be re-proven using graphical calculus in TC?
• RQ5 What conditions ensure that the set {ǫJ_I}I,J∈Irr(C) forms a complete set of orthogonal primitive idempotents in TC?

Key findings

• The Yoneda embedding ¥: TC → RT C is an idempotent completion, so RT C ≅ Z(C), establishing that TC provides a graphical model for Z(C).
• The idempotent ǫτ = 1/d(C) ⊗S d(S) · (X →S →X) in TC corresponds precisely to the half-braiding τ on X, realizing the image of (X, τ) ∈ Z(C) under the Yoneda embedding.
• The functor Φ: C ⊠ C → Z(C) is fully faithful, as HomT C(ǫYX, ǫBA) ≅ HomC(X,A) ⊗ HomC(Y,B) via the map f⊗g ↦ ǫBA ◦ (f⊗g) ◦ ǫYX.
• The set {ǫJ_I}I,J∈Irr(C) forms a complete set of orthogonal primitive idempotents in TC, as shown by dimension counting and the killing ring lemma.
• The composition map ∑IJ HomT C(X, ǫJ_I) ⊗ HomT C(ǫJ_I, Y) → HomT C(X,Y) is an isomorphism, proving essential surjectivity of Φ and hence the equivalence Z(C) ≅ C ⊠ C when C is modular.
• The converse holds: if C is not modular, then Φ fails to be an equivalence, as shown by the existence of non-trivial morphisms between ǫI_I∨ and ǫ1_1, implying degeneracy of the S-matrix.