Related papers: On Hopf Algebras and Their Generalizations
We show that if a finite dimensional Hopf algebra over ${\bf C}$ has a basis such that all the structure constants are non-negative, then the Hopf algebra must be given by a finite group $G$ and a factorization $G=G_+G_-$ into two…
This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric…
We say that a Hopf algebra H is semicocommutative if the right adjoint coaction factorizes through the tensor product of H with the center of H. For instance the commutative and the cocommutative Hopf algebras are semicocommutative. The…
Rational Hopf algebras (certain quasitriangular weak quasi-Hopf $^*$-algebras) are expected to describe the quantum symmetry of rational field theories. In this paper methods are developped which allow for a classification of all rational…
By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.
Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal…
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…
Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic…
In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a…
We introduce the notion of partial representation of a weak Hopf algebra. We present the universal algebra $H_{par}^w$, which factorizes these partial representations by algebra morphisms. Also, it is shown that $\Hp$ is isomorphic to a…
We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra ${\mathcal G}$ by adding a new generator $J$ satisfying $J^m=J$ for some integer $m$. We denote this algebra by $wU_q^{\tau}({\mathcal G})$. This algebra…
The notion of inner linear Hopf algebra is a generalization of the notion of discrete linear group. In this paper, we prove two general results that enable us to enlarge the class of Hopf algebras that are known to be inner linear: the…
We show that every partial representation of a connected Hopf algebra is global. Some interesting classes of partial representations of smash product Hopf algebras are studied, and a description of the partial "Hopf" algebra if the first…
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 study Sorkin's proposal of a generalization of quantum mechanics and find that the theories proposed derive their probabilities from $k$-th order polynomials in additive measures, in the same way that quantum mechanics uses a probability…
We develop Hopf-Galois theory for weak Hopf algebras, and recover analogs of classical results for Hopf algebras. Our methods are based on the recently introduced Galois theory for corings. We focus on the situatation where the weak Hopf…
A new quantization of groupoids under the name of \times-Hopf coalgebras is introduced. We develop a Hopf cyclic theory with coefficients in stable-anti-Yetter-Drinfeld modules for \times-Hopf coalgebras. We use \times-Hopf coalgebras to…
We give the explicit construction of the product of an arbitrary family of coalgebras, bialgebras and Hopf algebras: it turns out that the product of an arbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of a family…
This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called {\bf relatively Hopfian groups} and {\bf weakly Hopfian groups}, and establishing some their crucial properties and…
We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the…