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july 2020 Reconstruction of tensor categories from their structure invariants
Hui-Xiang Chen, Yinhuo Zhang
Bull. Belg. Math. Soc. Simon Stevin 27(2): 245-279 (july 2020). DOI: 10.36045/bbms/1594346417

Abstract

In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $\mathbb F$. Given a tensor category $\mathcal{C}$, we have two structure invariants of $\mathcal{C}$: the Green ring (or the representation ring) $r(\mathcal{C})$ and the Auslander algebra $A(\mathcal{C})$ of $\mathcal{C}$. We show that a Krull-Schmit abelian tensor category $\mathcal{C}$ of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of $\mathcal{C}$. In fact, we can reconstruct the tensor category $\mathcal{C}$ from its two invariants and the associator system. More general, given a quadruple $(R, A, \phi, a)$ satisfying certain conditions, where $R$ is a $\mathbb{Z}_+$-ring of rank $n$, $A$ is a finite dimensional $\mathbb F$-algebra with a complete set of $n$ primitive orthogonal idempotents, $\phi$ is an algebra map from $A\otimes_{\mathbb F}A$ to an algebra $M(R, A)$ constructed from $A$ and $R$, and $a=\{a_{i,j,l}|1\leqslant i,j,l\leqslant n\}$ is a family of ``invertible" matrices over $A$, we can construct a Krull-Schmidt and abelian tensor category $\mathcal C$ over $\mathbb{F}$ such that $R$ is the Green ring of $\mathcal C$ and $A$ is the Auslander algebra of $\mathcal C$. In this case, $\mathcal C$ has finitely many indecomposable objects (up to isomorphisms) and finite dimensional Hom-spaces. Moreover, we will give a necessary and sufficient condition for such two tensor categories to be tensor equivalent.

Citation

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Hui-Xiang Chen. Yinhuo Zhang. "Reconstruction of tensor categories from their structure invariants." Bull. Belg. Math. Soc. Simon Stevin 27 (2) 245 - 279, july 2020. https://doi.org/10.36045/bbms/1594346417

Information

Published: july 2020
First available in Project Euclid: 10 July 2020

zbMATH: 07242768
MathSciNet: MR4121373
Digital Object Identifier: 10.36045/bbms/1594346417

Subjects:
Primary: 16T05, 18D10

Rights: Copyright © 2020 The Belgian Mathematical Society

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Vol.27 • No. 2 • july 2020
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