[Paper Review] Planar algebras in braided tensor categories

André Henriques, David Penneys+1 more|arXiv (Cornell University)|Jul 20, 2016

Algebraic structures and combinatorial models4 citations

TL;DR

This paper introduces anchored planar algebras as a generalization of Jones' planar algebras internalized within a pivotal braided tensor category C. By equipping planar tangles with anchor lines that encode tensor order, the authors establish an equivalence between anchored planar algebras in C and pivotal module tensor categories over C equipped with a self-dual generator, extending the classical classification of planar algebras to higher categorical settings with full categorical equivalence.

ABSTRACT

We generalize Jones' planar algebras by internalising the notion to a pivotal braided tensor category $\mathcal{C}$. To formulate the notion, the planar tangles are now equipped with additional `anchor lines' which connect the inner circles to the outer circle. We call the resulting notion an anchored planar algebra. If we restrict to the case when $\mathcal{C}$ is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in $\mathcal{C}$ and pivotal module tensor categories over $\mathcal{C}$ equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras.

Motivation & Objective

• To generalize Jones' planar algebras from vector spaces to arbitrary pivotal braided tensor categories C.
• To define a new structure—anchored planar algebras—by introducing anchor lines that track tensor order in planar tangles.
• To establish a categorical equivalence between anchored planar algebras in C and pivotal module tensor categories over C with a self-dual generator.
• To extend the classical classification of planar algebras to a full categorical equivalence, previously unformulated in the literature.
• To provide a systematic construction of examples of anchored planar algebras via module tensor categories and vice versa.

Proposed method

• Internalize planar algebras by defining anchored planar tangles with red anchor lines connecting inner and outer circles, which determine the order of tensor factors.
• Define anchored planar algebras as functors from the anchored planar operad to a pivotal braided tensor category C, assigning morphisms to tangles.
• Use the categorified trace construction from prior work to relate module tensor categories over C to anchored planar algebras.
• Construct a functor from module tensor categories to anchored planar algebras using the tube string calculus and ribbon braid group actions.
• Prove that the assignment of tangles to morphisms is compatible with braid group actions and isotopy via the tube string diagram calculus.
• Establish a fully faithful and essentially surjective correspondence between the category of anchored planar algebras and the category of pivotal module tensor categories with self-dual generators.

Experimental results

Research questions

• RQ1 How can planar algebras be generalized to internalize within a pivotal braided tensor category C rather than just the category of vector spaces?
• RQ2 What structure on a module tensor category over C corresponds precisely to an anchored planar algebra in C?
• RQ3 Is there a categorical equivalence between anchored planar algebras in C and a specific class of module tensor categories over C?
• RQ4 How does the presence of anchor lines in tangles affect the assignment of morphisms and the composition rules in the planar algebra?
• RQ5 Can the classical classification of planar algebras via pivotal categories and self-dual generators be extended to a full categorical equivalence in the internalized setting?

Key findings

• Anchored planar algebras are defined by equipping planar tangles with anchor lines that determine the order of tensor factors, generalizing standard planar algebras to any pivotal braided tensor category C.
• The paper establishes a categorical equivalence between anchored planar algebras in C and pivotal module tensor categories over C equipped with a self-dual generator.
• The equivalence is constructed via a functor from module tensor categories to anchored planar algebras using the tube string calculus and categorified trace, with inverse construction via reconstruction of the module category from the planar algebra.
• The assignment of morphisms to tangles is compatible with the action of the ribbon braid group, ensuring invariance under isotopy.
• The main result generalizes the classical classification of planar algebras (via (D,X) pairs) to a full categorical equivalence, a formulation not previously available in the literature.
• The framework allows for the systematic construction of examples of anchored planar algebras from module tensor categories and vice versa, including via the tube string diagram calculus and braid group actions.