Related papers: Construction of factorizable Hopf algebras

We formulate the generation of finite dimensional pointed Hopf algebras by group-like elements and skew-primitives in geometric terms. This is done through a more general study of connected and coconnected Hopf algebras inside a braided…

We describe a method for quantization of Poisson Hopf algebras in $\mathbb Q$-linear symmetric monoidal categories. It is compatible with tensor products and can also be used to produce braided Hopf algebras. The main idea comes from the…

The aim of this paper is to construct a new braided $T$-category via the generalized Yetter-Drinfel'd modules and Drinfel'd codouble over Hopf algebra, an approach different from that proposed by Panaite and Staic \cite{PS}. Moreover, in…

We introduce and study a class of Hopf algebras $H(G, \chi, \eta, b, c, \beta)$ which are two-step Ore extensions of a group algebra $\mathbb{K}[G]$. This construction unifies and generalizes some known families of Hopf algebras such as…

One of the most fundamental problems in the theory of finite- dimensional Hopf algebras is their classification over an algebraically closed field k of characteristic 0. This problem is extremely difficult, hence people restrict it to…

Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We…

In a previous paper the authors constructed a class of quasi-Hopf algebras $D^{\omega}(G, A)$ associated to a finite group $G$, generalizing the twisted quantum double construction. We gave necessary and sufficient conditions, cohomological…

We classify the ribbon structures of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$. Our result generalizes Kauffman and Radford's classification result of the ribbon elements of the Drinfeld double…

This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras…

We present characterizations of braided co-Frobenius Hopf algebras in the braided tensor category of Yetter-Drinfeld modules over a Hopf algebra extending those already known for co-Frobenius Hopf algebras.

Let $\mathcal{C}$ be a finite braided multitensor category. Let $B$ be Majid's automorphism braided group of $\mathcal{C}$, then $B$ is a cocommutative Hopf algebra in $\mathcal{C}$. We show that the center of $\mathcal{C}$ is isomorphic to…

In arXiv:2503.19532 new examples of ribbon Hopf algebras based on the construction due to Nenciu were presented. This piece serves as a sequel where we study the representation theory of these new examples of ribbon Hopf algebras. We…

Inspired by the study of vertex operator algebra extensions, we answer the question of when the category of local modules over a commutative exact algebra in a braided finite tensor category is a (non-semisimple) modular tensor category.…

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 $nm^{2}$-dimensional quantum groups $T_{nm^{2}}^…

Starting from a Hopf algebra endowed with an action of a group G by Hopf automorphisms, we construct (by a twisted double method) a quasitriangular Hopf G-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular…

We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories -- and consequently, semi-simple Hopf algebras -- of dimension $pq^2$, where $p$ and $q$ are distinct primes.…

Doplicher and Roberts originally posed the problem of extending their duality theory for compact groups and field reconstruction to theories admitting braided symmetry. In this paper, we address this problem for the Wess-Zumino-Witten model…

We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects…

A family of algebra maps $H\to A_i$ whose common domain is a Hopf algebra is said to be jointly inner faithful if it does not factor simultaneously through a proper Hopf quotient of $H$. We show that tensor and free products of jointly…

We find a formula to compute the number of the generators, which generate the $n$-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight $n$. Applying Hopf algebra of rooted trees,…