# Exact factorizations and extensions of fusion categories

@article{Gelaki2016ExactFA,
title={Exact factorizations and extensions of fusion categories},
author={Shlomo Gelaki},
journal={arXiv: Quantum Algebra},
year={2016}
}
• S. Gelaki
• Published 4 March 2016
• Mathematics
• arXiv: Quantum Algebra
14 Citations

### A class of prime fusion categories of dimension $2^N$

• Mathematics
• 2019
We study a class of strictly weakly integral fusion categories $\mathfrak{I}_{N, \zeta}$, where $N \geq 1$ is a natural number and $\zeta$ is a $2^N$th root of unity, that we call $N$-Ising fusion

### Hopf Algebras which Factorize through the Taft Algebra Tm2(q) and the Group Hopf Algebra K[Cn]

We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra $T_{m^{2}}(q)$ and the group Hopf algebra $K[C_{n}]$: they are

### Algebraic structures in group-theoretical fusion categories

• Mathematics
• 2020
It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable,

### Extensions of tensor categories by finite group fusion categories

• S. Natale
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2019
Abstract We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual

### Exact factorizations and extensions of finite tensor categories

• Mathematics
• 2022
We extend [G1] to the nonsemisimple case. We define and study exact factorizations B = A • C of a finite tensor category B into a product of two tensor subcategories A ,C ⊂ B, and relate exact

### Subalgebras of etale algebras and fusion subcategories

. In [7, Rem. 3.4] the authors asked the question if any étale subalgebra of an étale algebra in a braided fusion category is also étale. We give a positive answer to this question if the braided

### The factorization problem for Jordan algebras: applications

• Mathematics
Collectanea Mathematica
• 2022
. We investigate the factorization problem as well as the classifying complements problem in the setting of Jordan algebras. Matched pairs of Jordan algebras and the corresponding bicrossed products

### Slightly trivial extensions of a fusion category

We introduce and study the notion of slightly trivial extensions of a fusion category which can be viewed as the first level of complexity of extensions. We also provide two examples of slightly

## References

SHOWING 1-10 OF 19 REFERENCES

### Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories

Abstract.Given an action of a finite group G on a fusion category $${\mathcal{C}}$$ we give a criterion for the category of G-equivariant objects in $${\mathcal{C}}$$ to be group-theoretical, i.e.,

### Central exact sequences of tensor categories, equivariantization and applications

• Mathematics
• 2011
We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization

### Classifying complements for groups. Applications

• Mathematics
• 2012
Let $A \leq G$ be a subgroup of a group $G$. An $A$-complement of $G$ is a subgroup $H$ of $G$ such that $G = A H$ and $A \cap H = \{1\}$. The \emph{classifying complements problem} asks for the

### Classifying Bicrossed Products of Hopf Algebras

• Mathematics
• 2014
Let A and H be two Hopf algebras. We shall classify up to an isomorphism that stabilizes A all Hopf algebras E that factorize through A and H by a cohomological type object ${\mathcal H}^{2} (A, H)$.

### Exact sequences of tensor categories

• Mathematics
• 2010
We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as

### On fusion categories

• Mathematics
• 2002
Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show

### Module categories over equivariantized tensor categories

• Mathematics
• 2014
For a finite tensor category $\mathcal C$ and a Hopf monad $T:\mathcal C\to \mathcal C$ satisfying certain conditions we describe exact indecomposable left $\mathcal C^T$-module categories in terms