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We construct multi-brace cotensor Hopf algebras with bosonizations of quantum multi-brace algebras as examples. Quantum quasi-symmetric algebras are then obtained by taking particular initial data; this allows us to realize the whole…
We obtain the double factorization of braided bialgebras or braided Hopf algebras, give relation among integrals and semisimplicity of braided Hopf algebra and its factors.
Let $ Aut_{mHH}(H)$ denote a set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel S. and Goyvaerts I. (Commun. Algebra 39, 2216-2240, 2011) and let $G$ be a crossed product group $…
Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is…
We show that for a braided Hopf algebra in the category of comodules over a cosemisimple coquasitriangular Hopf algebra, the Hochschild cohomological dimension, the left and right global dimensions and the projective dimensions of the…
Firstly, we introduce a class of new algebraic systems which generalize Hopf quasigroups and Hopf $\pi-$algebras called $Q$-graded Hopf quasigroups, and research some properties of them. Secondly, we define the representations of $Q$-graded…
We show that in characteristic zero all irreducible symmetrically braided Hopf algebras are of PBW type. Consequently, we obtain conditions for a braided Hopf algebra to be of PBW type as module over a braided Hopf subalgebra containing the…
Generalising a result for Hopf algebras, we not only define the four possible types of Hopf modules in the bialgebroid setting but also yield the notion of two-sided two-cosided Hopf modules, also known as Hopf bimodules or tetramodules, in…
We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of…
We prove that any fusion category over $\mathbb{C}$ with exactly one non-invertible simple object is spherical. Furthermore, we classify all such categories that come equipped with a braiding.
We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories…
A quasi-Hopf algebra $H$ can be seen as a commutative algebra $A$ in the centre $\mathcal Z(H-Mod)$ of $H-Mod$. We show that the category of $A$-modules in $\mathcal Z(H-Mod)$ is equivalent (as a monoidal category) to $H-Mod$. This can be…
In this paper we define and study the algebraic conterpart of sovereign monoidal categories : cosovereign Hopf algebras.
A fundamental problem in the theory of Hopf algebras is the classification and explicit construction of finite-dimensional quasitriangular Hopf algebras over C. These Hopf algebras constitute a very important class of Hopf algebras,…
We provide a parameterization of all fusion subcategories of the equivariantization by a group action on a fusion category. As applications, we classify the Hopf subalgebras of a family of semisimple Hopf algebras of Kac-Paljutkin type and…
The main properties of the crossed product in the category of Hopf algebras are investigated. Let $A$ and $H$ be two Hopf algebras connected by two morphism of coalgebras $\triangleright : H\ot A \to A$, $f:H\ot H\to A$. The crossed product…
Associated to each subset $J$ of the nodes $I$ of a Dynkin diagram is a triangular decomposition of the corresponding Lie algebra $\mathfrak{g}$ into three subalgebras $\widetilde{\mathfrak{g}_{J}}$ (generated by $e_{j}$, $f_{j}$ for $j\in…
We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to g-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these g-quasi-Poisson…
We define a family of Hopf algebra objects, $H$, in the braided category of $\mathbb{Z}_n$-modules (known as anyonic vector spaces), for which the property $\psi^2_{H\otimes H}=id_{H\otimes H}$ holds. We will show that these anyonic Hopf…
Given a crossed module $\chi$, we introduce Hopf $\chi$-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of…