Related papers: Monadic cointegrals and applications to quasi-Hopf…

We introduce the notion of `bar category' by which we mean a monoidal category equipped with additional structure formalising the notion of complex conjugation. Examples of our theory include bimodules over a $*$-algebra, modules over a…

Braided Morita invariants of finite-dimensional semisimple and cosemisimple Hopf algebras with braidings are constructed by refining the polynomial invariants introduced by the author. The invariants are computed for the duals of Suzuki's…

We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the…

We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the…

Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra theory are not…

The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There…

The faithful quasi-dual $H^d$ and strict quasi-dual $H^{d'}$ of an infinite braided Hopf algebra $H$ are introduced and it is proved that every strict quasi-dual $H^{d'}$ is an $H$-Hopf module. The connection between the integrals and the…

In a previous work \cite{AS2} we showed how to attach to a pointed Hopf algebra A with coradical $\k\Gamma$, a braided strictly graded Hopf algebra R in the category $_{\Gamma}^{\Gamma}\Cal{YD}$ of Yetter-Drinfeld modules over $\Gamma$. In…

We classify finite-dimensional complex pointed Hopf algebra with group of group-like elements isomorphic to A_5. We show that any pointed Hopf algebra with infinitesimal braiding associated with the conjugacy class of $\pi$ \in $A_n$ is…

Due to the work of Shimizu (2019), various nondegeneracy conditions for braided finite tensor categories are equivalent. This theory is partially extended to braided module categories here. We introduce when a braided module category is…

It is shown in math.QA/0310253 that a finite dimensional quasi-Hopf algebra over the complex numbers with radical of codimension 2 is twist equivalent to a Nichols Hopf algebra, or to a lifting of one of four special quasi-Hopf algebras of…

Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor…

Hopf (bi-)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra…

We prove the graded braided commutativity of the Hochschild cohomology of $A$ with trivial coefficients, where $A$ is a braided Hopf algebra in the category of Yetter-Drinfeld modules over the group algebra of an abelian group, under some…

Recall that a triangular Hopf algebra A is said to have the Chevalley property if the tensor product of any two simple A-modules is semisimple, or, equivalently, if the radical of A is a Hopf ideal. There are two reasons to study this class…

For a braided finite tensor category $\mathcal{C}$ with unit object $1 \in \mathcal{C}$, Lyubashenko considered a certain Hopf algebra $\mathbb{F} \in \mathcal{C}$ endowed with a Hopf pairing $\omega: \mathbb{F} \otimes \mathbb{F} \to 1$ to…

For each braided category $\mathcal{C}$ we show that, under mild hypotheses, there is an associated category of "half braided algebras" and their bimodules internal to $\mathcal{C}$ which is not only monoidal but even braided and balanced.…

Let $A$ be a ring and $\M_A$ the category of $A$-modules. It is well known in module theory that for any $A $-bimodule $B$, $B$ is an $A$-ring if and only if the functor $-\otimes_A B: \M_A\to \M_A$ is a monad (or triple). Similarly, an $A…

Let p be a prime, and denote the class of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p, by RG(p). The purpose of this paper is to continue the structure theory of finite dimensional…

A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known…