Related papers: Bosonization for dual quasi-bialgebras and preanti…

It is known that a dual quasi-bialgebra with antipode $H$, i.e. a dual quasi-Hopf algebra, fulfils a fundamental theorem for right dual quasi-Hopf $H$-bicomodules. The converse in general is not true. We prove that, for a dual…

Let $H$ be a quasitriangular quasi-Hopf algebra, we construct a braided group $\underline{H}$ in the quasiassociative category of left $H$-modules. Conversely, given any braided group $B$ in this category, we construct a quasi-Hopf algebra…

In this paper we continue the investigation started in [A.M.St.-Small], dealing with bialgebras $A$ with an $H$-bilinear coalgebra projection over an arbitrary subbialgebra $H$ with antipode. These bialgebras can be described as deformed…

Double-bosonisation associates to a braided group in the category of modules of a quantum group, a new quantum group. We announce the semiclassical version of this inductive construction.

We study the relationship between general dynamical Poisson groupoids and Lie quasi-bialgebras. For a class of Lie quasi-bialgebras naturally compatible with a reductive decomposition, we extend the description of the moduli space of…

Let $A$ be a Hopf algebra over a field $K$ of characteristic zero such that its coradical $H$ is a finite dimensional sub-Hopf algebra. Our main theorem shows that there is a gauge transformation $\zeta $ on $A$ such that $A^{\zeta}\cong…

The purpose of this paper is to give a general survey of Hom-bialgebras, which are bialgebra-type structures where the identities are twisted by a morphism, and to extend the concept of quasi-bialgebra to Hom-setting. We provide some key…

We prove that a finite-dimensional Hopf algebra with the dual Chevalley Property over a field of characteristic zero is quasi-isomorphic to a Radford-Majid bosonization whenever the third Hochschild cohomology group in the category of…

Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any Lie quasi-bialgebra structure of finite-dimensional (G, \mu, \gamma ,\phi ?), correspond one Lie algebra structure on D = G\oplus G*, called…

The aim of this paper is to define and study Yetter-Drinfeld modules over Hom-bialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a…

In this paper, the notions of quasi-triangular and factorizable dual pre-Poisson bialgebras are introduced. A factorizable dual pre-Poisson bialgebra induces a factorization of the underlying dual pre-Poisson algebra, and the double of any…

The classical Stone duality associates to each Boolean algebra a topological space consisting of ultrafilters. Lawson's generalisation constructs a dual equivalence of categories of Boolean inverse $\land$-semigroups and Hausdorff ample…

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with…

Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful…

This paper is a coalgebra version of arXiv:1703.04266 and a sequel to arXiv:1607.03066. We present the definition of a pseudo-dualizing complex of bicomodules over a pair of coassociative coalgebras $\mathcal C$ and $\mathcal D$. For any…

The Structure Theorem for Hopf modules states that if a bialgebra $H$ is a Hopf algebra (i.e. it is endowed with a so-called antipode) then every Hopf module $M$ is of the form ${M}^{\mathrm{co}{H}}\otimes H$, where ${M}^{\mathrm{co}{H}}$…

We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that…

Biserial algebras are a classical class in the representation theory of algebras, generalizing Nakayama algebras. They were further generalized by Green and Schroll to multiserial algebras, which share many structural properties with…

Up to Morita equivalence, every quasi-hereditary algebra is the dual algebra of a directed bocs or coring. From the bocs, an exact Borel subalgebra is obtained. In this paper a characterisation of exact Borel subalgebras arising in this way…

This paper is devoted to the presentation of combinatorial bialgebras whose coproduct is defined with the help of a commutative semigroup. We consider this setting in order to give a general framework which admits as special cases the…