We present an overview of the notions of exact sequences of Hopf algebras and tensor categories and their connections. We also present some examples illustrating their main features; these include simple fusion categories and a natural question regar
ding composition series of finite tensor categories.
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 categories. An $N$-Ising fusion category has Frobenius-Pe
rron dimension $2^{N+1}$ and is a graded extension of a pointed fusion category of rank 2 by the cyclic group of order $mathbb Z_{2^N}$. We show that every braided $N$-Ising fusion category is prime and also that there exists a slightly degenerate $N$-Ising braided fusion category for all $N > 2$. We also prove a structure result for braided extensions of a rank 2 pointed fusion category in terms of braided $N$-Ising fusion categories.
We study exact sequences of finite tensor categories of the form $Rep G to C to D$, where $G$ is a finite group. We show that, under suitable assumptions, there exists a group $Gamma$ and mutual actions by permutations $rhd: Gamma times G to G$ and $
lhd: Gamma times G to Gamma$ that make $(G, Gamma)$ into matched pair of groups endowed with a natural crossed action on $D$ such that $C$ is equivalent to a certain associated crossed extension $D^{(G, Gamma)}$ of $D$. Dually, we show that an exact sequence of finite tensor categories $vect_G to C to D$ induces an $Aut(G)$-grading on $C$ whose neutral homogeneous component is a $(Z(G), Gamma)$-crossed extension of a tensor subcategory of $D$. As an application we prove that such extensions $C$ of $D$ are weakly group-theoretical fusion categories if and only if $D$ is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.
We show that the core of a weakly group-theoretical braided fusion category $C$ is equivalent as a braided fusion category to a tensor product $B boxtimes D$, where $D$ is a pointed weakly anisotropic braided fusion category, and $B cong vect$ or $B$
is an Ising braided category. In particular, if $C$ is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Frobenius-Perron dimension at most 2 is necessarily group-theoretical.
Let $C$ be a modular category of Frobenius-Perron dimension $dq^n$, where $q$ is a prime number and $d$ is a square-free integer. We show that if $q>2$ then $C$ is integral and nilpotent. In particular, $C$ is group-theoretical. In the general case,
we describe the structure of $C$ in terms of equivariantizations of group-crossed braided fusion categories.
Let $p$ and $q$ be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $pq^2$. We obtain that the $p+1$ non-isomorphic
self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac-Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.
We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${mathcal C}$ to be equivalent. This concludes the classification of such module categories.
We address the question whether the condition on a fusion category being solvable or not is determined by its fusion rules. We prove that the answer is affirmative for some families of non-solvable examples arising from representations of semisimple
Hopf algebras associated to exact factorizations of the symmetric and alternating groups. In the context of spherical fusion categories, we also consider the invariant provided by the $S$-matrix of the Drinfeld center and show that this invariant does determine the solvability of a fusion category provided it is group-theoretical.
We prove a version of the Jordan-H older theorem in the context of weakly group-theoretical fusion categories. This allows us to introduce the composition factors and the length of such a fusion category C, which are in fact Morita invariants of C.
We show that a Jordan-Holder theorem holds for appropriately defined composition series of finite dimensional Hopf algebras. This answers an open question of N. Andruskiewitsch. In the course of our proof we establish analogues of the Noether isomorp
hism theorems of group theory for arbitrary Hopf algebras under certain faithful (co)flatness assumptions. As an application, we prove an analogue of Zassenhaus butterfly lemma for finite dimensional Hopf algebras. We then use these results to show that a Jordan-Holder theorem holds as well for lower and upper composition series, even though the factors of such series may be not simple as Hopf algebras.
