Related papers: The antipode of a dual quasi-Hopf algebra with non…

We classify finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradcial is isomorphic to the smallest non-pointed basic Hopf algebra, under the assumption that the diagrams are strictly…

We study integral theory for left (or right) Hopf left bialgebroids. Contrary to Hopf algebroids, the latter ones don't necessary have an antipode $S$ but, for any element $u$, the elements $u_{(1)} \otimes S(u_{(2)})$ (or $u_{(2)}\otimes…

We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an…

In this paper, we introduce and study the notion of cointegrals in a weak multiplier Hopf algebras $(A, \Delta)$. A cointegral is a non-zero element $h$ in the multiplier algebra $M(A)$ such that $ah=\v_t(a)h$ for any $a\in A$. When $A$ has…

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on algebras. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the…

Let A be a cogroup in the category of connected graded algebras over a commutative ring R. Let nu denote the inverse of A and chi the antipode of the underlying Hopf algebra of A. We clarify the differences and similarities of nu and chi,…

Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic zero, where p, q are odd primes with p < q < 4p+12. We prove that H is semisimple and thus isomorphic to a group algebra, or the dual of a group…

We introduce a general class of combinatorial objects, which we call \emph{multi-complexes}, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of…

Let $H$ be a $\times$-bialgebra in the sense of Takeuchi. We show that if $H$ is $\times$-Hopf, and if $H$ fulfills the finiteness condition necessary to define its skew dual $H^\vee$, then the coopposite of the latter is $\times$-Hopf as…

The Hopf dual $H^\circ$ of any Poisson Hopf algebra $H$ is proved to be a co-Poisson Hopf algebra provided $H$ is noetherian. Without noetherian assumption, it is not true in general. There is no nontrivial Poisson Hopf structure on the…

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as…

We introduce the notion of a homological integral for an infinite-dimensional weak Hopf algebra and use the homological integral to prove several structure theorems. For example, we prove that the Artin--Schelter property and the Van den…

In this paper, we obtain a canonical central element $\nu_H$ for each semi-simple quasi-Hopf algebra $H$ over any field $k$ and prove that $\nu_H$ is invariant under gauge transformations. We show that if $k$ is algebraically closed of…

In this paper we introduce the notion of a Hopf C*-algebra and construct the counit and antipode. A Hopf C*-algebra is a C*-algebra with comultiplication satisfying some extra condition which makes possible the construction of the counit…

For an infinite dimensional Hopf quasigroup, if the faithful integral exists, then it is unique up to scalar. Base on the faithful integrals, we construct the integral dual of a class of infinite dimensional Hopf quasigroups, and show that…

We find a self-dual noncommutative and noncocommutative Hopf algebra acting as a universal symmetry on the modules over inner Frobenius algebras of modular categories (as used in two dimensional boundary conformal field theory) similar to…

We introduce quasi-Hopf $*$-algebras i.e. quasi-Hopf algebras equipped with a conjugation (star) operation. The definition of quasi-Hopf $*$-algebras proposed ensures that the class of quasi-Hopf $*$-algebras is closed under twisting and…

We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf…

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf…

We identify a collection of primitive elements generating the Hopf algebra NCSym of symmetric functions in noncommuting variables and give a combinatorial formula for the antipode.