The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second group is the set of formal diffeomorphisms with the group law being composition of series. The motivation to introduce these Hopf algebras comes ...

In loving memory of my mother " A mathematician is a machine for converting coffee into theorems. " —Alfréd Rényi " A comathematician, by categorical duality, is a machine for converting cotheorems into ffee. " —anonymous Preface Hopf algebras are a relatively new concept in algebra, first encountered by their namesake Heinz Hopf in 1941 in the field of algebraic topology. In the 1960s, study o...

We provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions.

Some new algebraic structures related to the coloured Yang-Baxter equation, and termed coloured Hopf algebras, are reviewed. Coloured quantum universal enveloping algebras of Lie algebras are defined in this context. An extension to the coloured graded Yang-Baxter equation and coloured Hopf superalgebras is also presented. The coloured two-parameter quantum universal enveloping algebra of gl(1/...

Let (H, σ) be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras, we define Hσ , a sub-Hopf algebra of H, the finite dual of H. Using the generalized quantum double construction and the theory of Hopf algebras with a projection, we...

Preface The term " quantum group " was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics and physics. The development of Hopf algebras can be divided into five stages. The first...

Let m ≥ 1 be a positive integer. We show that, over an algebraically closed field of characteristic zero, there exists cosemisimple Hopf algebras having an antipode of order 2m. We also discuss the Schur indicator for such Hopf algebras.

We extend the notion of dually conjugate Hopf (super)algebras to the coloured Hopf (super)algebras Hc that we recently introduced. We show that if the standard Hopf (super)algebras Hq that are the building blocks of Hc have Hopf duals H∗ q , then the latter may be used to construct coloured Hopf duals Hc∗, endowed with coloured algebra and antipode maps, but with a standard coalgebraic structur...

We prove the existence of a basis of Poincaré-Birkhoff-Witt type for braided Hopf algebras R generated by a braided subspace V ⊂ P (R) if the braiding on V fulfils a triangularity condition. We apply our result to pointed Hopf algebras with abelian coradical and to Nichols algebras of low dimensional simple Uq(sl2)-modules.

We consider an inductive scheme for quantized enveloping algebras, arising from certain inclusions of the associated root data. These inclusions determine an algebra-subalgebra pair with the subalgebra also a quantized enveloping algebra, and we want to understand the structure of the “difference” between the algebra and the subalgebra. Our point of view treats the background field and quantiza...