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Towards reconstruction of finite tensor categories

Mitchell Jubeir, Zhenghan Wang

TL;DR

The work addresses reconstructing finite tensor categories from finite data by focusing on the projective subcategory as a complete invariant. It develops a rigorous reconstruction framework for non unital and abelian settings and demonstrates the approach on Nichols Hopf algebras, especially the eight dimensional $K_2$, through a detailed analysis of Green rings and tensor ideals. The key contributions include proving that the full projective subcategory uniquely determines the category up to equivalence, constructing categorical reconstruction from projectives, and explicitly computing the Green ring $r(K_2)$ along with its tensor ideals via connections to the Drinfeld double $DK_1$. The results illuminate the complexity arising from abundant non projective indecomposables and motivate further classification of finite tensor categories by their projective data, with potential applications to non semisimple modular structures and topological invariants.

Abstract

We take a first step towards a reconstruction of finite tensor categories using finitely many $F$-matrices. The goal is to reconstruct a finite tensor category from its projective ideal. Here we set up the framework for an important concrete example--the $8$-dimensional Nicholas Hopf algebra $K_2$. Of particular importance is to determine its Green ring and tensor ideals. The Hopf algebra $K_2$ allows the recovery of $(2+1)$-dimensional Seiberg-Witten TQFT from Hennings TQFT based on $K_2$. This powerful result convinced us that it is interesting to study the Green ring of $K_2$ and its tensor ideals in more detail. Our results clearly illustrate the difficulties arisen from the proliferation of non-projective reducible indecomposable objects in finite tensor categories.

Towards reconstruction of finite tensor categories

TL;DR

The work addresses reconstructing finite tensor categories from finite data by focusing on the projective subcategory as a complete invariant. It develops a rigorous reconstruction framework for non unital and abelian settings and demonstrates the approach on Nichols Hopf algebras, especially the eight dimensional , through a detailed analysis of Green rings and tensor ideals. The key contributions include proving that the full projective subcategory uniquely determines the category up to equivalence, constructing categorical reconstruction from projectives, and explicitly computing the Green ring along with its tensor ideals via connections to the Drinfeld double . The results illuminate the complexity arising from abundant non projective indecomposables and motivate further classification of finite tensor categories by their projective data, with potential applications to non semisimple modular structures and topological invariants.

Abstract

We take a first step towards a reconstruction of finite tensor categories using finitely many -matrices. The goal is to reconstruct a finite tensor category from its projective ideal. Here we set up the framework for an important concrete example--the -dimensional Nicholas Hopf algebra . Of particular importance is to determine its Green ring and tensor ideals. The Hopf algebra allows the recovery of -dimensional Seiberg-Witten TQFT from Hennings TQFT based on . This powerful result convinced us that it is interesting to study the Green ring of and its tensor ideals in more detail. Our results clearly illustrate the difficulties arisen from the proliferation of non-projective reducible indecomposable objects in finite tensor categories.
Paper Structure (31 sections, 16 theorems, 45 equations)

This paper contains 31 sections, 16 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Non-unital tensor categories
  4. Fusion rings, tensor ideals, and projective Auslander algebras
  5. Green rings $r(\mathcal{C})$ and projective fusion rings $K_0(\mathcal{C})$
  6. Tensor ideals
  7. Projective Auslander algebras
  8. Nichols Hopf algebras, their projective fusion rings and Auslander algebras
  9. Nichols Hopf algebras $K_m$
  10. Finite tensor categories $\mathop{\mathrm{Rep}}\nolimits(K_m)$ and their projective ideals
  11. Projective Auslander algebras of $K_m$
  12. Subcategories as complete invariants and reconstruction
  13. Full projective subcategory as a complete invariant
  14. Full projective subcategory of an abelian category with enough projectives
  15. Quasi-dominated subcategory as a complete invariant
  16. ...and 16 more sections

Key Result

Proposition 2.1

The projective Auslander algebra of $\mathop{\mathrm{Rep}}\nolimits(K_m)$ is isomorphic as an algebra to $K_m$ for each $m\geq 1$.

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 24 more