Let A be a Hopf algebra and Γ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra Γ = Γ/J that contains Γ as its first order part. Corresponding to the three choices of the ideal J , we distinguish the ‘universal’ exterior algebra, the ‘second antisymmetrizer’ exterior algebra, and Woronowicz’ externa...

This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from Rn by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold S × S, conformally flat cylinders and tori and some conformally flat manifolds of gen...

Noncommutative (deformed, quantum) spaces are deformations of the usual commutative space-time. They depend on parameters, such that for certain values of parameters they become the usual space-time. The symmetry acting on them is given in terms of a deformed quantum group symmetry. In this work we discuss two special examples, the θ-deformed space and the κ-deformed space. In the case of the θ...

We introduce the concept of para-Hopf algebroid and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Para-Hopf algebroids are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, ...

We introduce the concept of extended Hopf algebras and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Extended Hopf algebras are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structu...

The concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasi-bicrossed products are constructed by means of weak Hopf skew-pairs of the weak Hopf algebras which are generalizations of the Hopf pairs introduced by Takeuchi. As a special case, the quantum double of a finite dimensional biperfect (noncocommutative) weak Hopf a...

Let T be an invertible transformation of a Borel space X with a-nite quasi-invariant measure m. One can single out the following natural types of behaviour of T on a measurable subset A X: 1) A is invariant, i.e. TA = A; 2) A is recurrent, i.e., for a.e. point x 2 A there is n = n(x) > 0 such that T n x 2 A; 3) A is wandering, i.e., all its translations are pairwise disjoint (so that a.e. point...