A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure of an infinite Hopf family of (super-) algebras is proposed. The structure of the infinite Hopf family is given explicitly on the level of L matrices. The relation with the Drinfeld current realization is established in the case of 4 × 4 R-matrices by studying the analogue of the Ding-Frenkel theor...

The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined. Introduction The purpose of this paper is to compare two previously defined invariants of 3-manifolds. Let A be a finite-dimensional Hopf algebra over a field F with antipode S. Then if S = 1 t...

Pseudo-Galois extensions are shown to be depth two extensions. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups or involutive Hopf algebras and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observ...

Let p, q be distinct prime numbers, and k an algebraically closed field of characteristic 0. Under certain restrictions on p, q, we discuss the structure of semisimple Hopf algebras of dimension pq. As an application, we obtain the structure theorems for semisimple Hopf algebras of dimension 9q over k. As a byproduct, we also prove that odd-dimensional semisimple Hopf algebras of dimension less...

– We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p > 17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius–Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. ...

Cyclic cohomology of Hopf algebras admitting a modular pair was first defined in [2] and further developed in [3] and [4] in the context of transverse geometry. Their results are followed by several papers computing Hopf cyclic (co)homology of certain Hopf algebras such as [15], [5] and [13]. In a series of papers [1], [10], [11], and [6] the authors developed a theory of cyclic (co)homology wh...

In this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to efficiently compute certain differential operators. Given data structures such as trees which can be multiplied, and which act as higherorder derivat...

The aim of this work is to discuss the concepts of degeneration, deformation and rigidity of Hopf algebras and to apply them to the geometric study of the varieties of Hopf algebras. The main result is the description of the ndimensional rigid Hopf algebras and the irreducible components for n = p, p is prime number, and for n < 14. This paper is organized as follows : first we present the basi...

Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a Hopf algebra. In this paper, we generalize this result to possibly infinite-dimensional algebras, with or without identity. We have to leave the setting of H...