A Hopf module is an A-module for an algebra A as well as a C-comodule for a coalgebra C, satisfying a suitable compatibility condition between the module and comodule structures. To formulate the compatibility condition one needs some kind of interaction between A and C. The most classical case arises when A = C =: H is a bialgebra. Many subsequent variants of this were unified independently by...

An algebraic deformation theory of module-algebras over a bialgebra is constructed. The cases of module-coalgebras, comodule-algebras, and comodule-coalgebras are also considered.

We establish a bijective correspondence between gauge equivalence classes of dynamical twists in a finite-dimensional Hopf algebra H based on a finite abelian group A and equivalence classes of pairs (K, {V λ } λ∈ b A), where K is an H-simple left H-comodule semisimple algebra and {V λ } λ∈ b A is a family of irreducible representations satisfying certain conditions. Our results generalize the ...

This is a continuation of previous study initiated by one us on nonlocal vertex bialgebras and smash product algebras. In this paper, we notion right $H$-comodule algebra for bialgebra $H$ give construction deformations algebras with structure compatible $H$-module structure. We also $\phi$-coordinated quasi modules As an example, family quantum deforming the associated to non-degenerate even l...

We establish a bijective correspondence between gauge equivalence classes of dynamical twists in a finite-dimensional Hopf algebra H based on a finite abelian group A and equivalence classes of pairs (K, {Vλ}λ∈Â), where K is an H-simple left H-comodule semisimple algebra and {Vλ}λ∈Â is a family of irreducible representations satisfying certain conditions. Our results generalize the results obta...

The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a review of the relevant techniques in categories of left modules, applications to the bimodule struc...

We prove First Fundamental Theorems of Coinvariant Theory for the standard coactions of the quantum groups Oq(GLt(K)) and Oq(SLt(K)) on the quantized algebra Oq(Mm,t(K)) ⊗ Oq(Mt,n(K)). (Here K is an arbitrary field and q an arbitrary nonzero scalar.) In both cases, the set of coinvariants is a subalgebra of Oq(Mm,t(K))⊗Oq(Mt,n(K)), which we identify. Introduction One of the highlights of classi...

Let H be a quasi-Hopf algebra. We show that any H-bimodule coalgebra C for which there exists an H-bimodule coalgebra morphism ν : C → H is isomorphic to what we will call a smash product coalgebra. To this end, we use an explicit monoidal equivalence between the category of two-sided two-cosided Hopf modules over H and the category of left Yetter-Drinfeld modules over H. This categorical metho...

We show that certain embeddable homogeneous spaces of a quantum group that do not correspond to a quantum subgroup still have the structure of quantum quotient spaces. We propose a construction of quantum fibre bundles on such spaces. The quantum plane and the general quantum two-spheres are discussed in detail. 0. Introduction A homogeneous space X of a Lie group G may be always identified wit...