[Paper Review] Characterizations of unimodular finite tensor categories

Kenichi Shimizu|arXiv (Cornell University)|Feb 14, 2014

Algebraic structures and combinatorial models25 references4 citations

TL;DR

This paper establishes multiple equivalent characterizations of unimodular finite tensor categories by analyzing the forgetful functor from the center of a category to the category itself. It proves that unimodularity is equivalent to the forgetful functor being Frobenius, and to the left and right adjoints preserving duality or mapping the unit object to self-dual objects, generalizing invariants for handlebody-links.

ABSTRACT

Let $\mathcal{C}$ be a finite tensor category, let $\mathcal{Z}(\mathcal{C})$ denote its center, and let $L$ and $R$ be a left and a right adjoint functor of the forgetful functor $U: \mathcal{Z}(\mathcal{C}) o \mathcal{C}$. We show that the following assertions are equivalent: (i) $\mathcal{C}$ is unimodular, (ii) $U$ is a Frobenius functor, (iii) $L$ preserves duality, (iv) $L(1)$ is self-dual, (v) $R$ preserves duality, and (vi) $R(1)$ is self-dual, where $1 \in \mathcal{C}$ is the unit object. Some other equivalent assertions are also given. As an application, we generalize Ishii and Masuoka's construction of an invariant of handlebody-links to unimodular finite tensor categories.

Motivation & Objective

• To identify and unify multiple equivalent conditions for a finite tensor category to be unimodular.
• To clarify the role of the forgetful functor from the center of a category to the category itself in determining unimodularity.
• To extend Ishii and Masuoka's handlebody-link invariant construction to the broader setting of unimodular finite tensor categories.
• To characterize unimodularity through properties of the left and right adjoints of the forgetful functor.

Proposed method

• Analyzing the forgetful functor $ U: \mathcal{Z}(\mathcal{C}) \to \mathcal{C} $ and its left and right adjoints $ L $ and $ R $.
• Establishing equivalence between unimodularity and $ U $ being a Frobenius functor.
• Proving that $ L $ preserves duality if and only if $ \mathcal{C} $ is unimodular.
• Showing that $ L(1) $ is self-dual precisely when $ \mathcal{C} $ is unimodular, where $ 1 $ is the unit object.
• Demonstrating that $ R $ preserves duality if and only if $ \mathcal{C} $ is unimodular.
• Using these equivalences to generalize an invariant for handlebody-links from modular categories to unimodular finite tensor categories.

Experimental results

Research questions

• RQ1 What conditions on a finite tensor category $ \mathcal{C} $ are equivalent to unimodularity?
• RQ2 How does the forgetful functor $ U: \mathcal{Z}(\mathcal{C}) \to \mathcal{C} $ relate to Frobenius functors in the context of unimodular categories?
• RQ3 In what way do the left and right adjoints $ L $ and $ R $ of $ U $ reflect the unimodularity of $ \mathcal{C} $?
• RQ4 Can the invariants of handlebody-links constructed by Ishii and Masuoka be extended beyond modular categories?
• RQ5 What is the significance of $ L(1) $ and $ R(1) $ being self-dual in relation to unimodularity?

Key findings

• Unimodularity of $ \mathcal{C} $ is equivalent to the forgetful functor $ U: \mathcal{Z}(\mathcal{C}) \to \mathcal{C} $ being a Frobenius functor.
• The left adjoint $ L $ preserves duality if and only if $ \mathcal{C} $ is unimodular.
• $ L(1) $ is self-dual if and only if $ \mathcal{C} $ is unimodular, where $ 1 $ is the unit object in $ \mathcal{C} $.
• The right adjoint $ R $ preserves duality if and only if $ \mathcal{C} $ is unimodular.
• $ R(1) $ is self-dual if and only if $ \mathcal{C} $ is unimodular.
• The construction of handlebody-link invariants by Ishii and Masuoka is generalized to unimodular finite tensor categories.