In the theory of dynamical Yang-Baxter equation, with any Hopf algebra H and a certain H-module and H-comodule algebra L (base algebra) one associates a monoidal category. Given an algebra A in that category, one can construct an associative algebra A⋊L, which is a generalization of the ordinary smash product when A is an ordinary H-algebra. We study this ”dynamical smash product” and its modul...

For a Hopf algebra H over a commutative ring k, the category MH of right Hopf modules is equivalent to the category Mk of k-modules, that is, the comparison functor −⊗k H : Mk → MH is an equivalence (Fundamental theorem of Hopf modules). This was proved by Larson and Sweedler via the notion of coinvariants McoH for any M ∈ MH . The coinvariants functor (−) coH : MH → Mk is right adjoint to the ...

Let $H$ be a bialgebra. $\sigma: H\otimes H\to A$ linear map, where $A$ is left $H$-comodule coalgebra, and an algebra with $H$-weak action $\triangleright$. $\tau: B$ $B$ right $\triangleleft$. In this paper, we improve the necessary conditions for two-sided crossed product $A\#^{\sigma} H~{^{\tau}\#} smash coproduct coalgebra $A\times H\times to form bialgebra (called double biproduct) such t...

This paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite Hopf algebras on commutative rings. Suppose that H is a finite dimensional Hopf algebra and A a commutative algebra, say over a field K. Let δ : A → A ⊗H be an algebra homomorphism which makes A into a right H-comodule. In this case A is called an H-comodule algebra. T...

We construct the group of H-Galois objects for a flat and cocommutative Hopf algebra in braided monoidal category with equalizers provided that certain assumption on braiding is fulfilled. show it subgroup BiGalois Schauenburg, prove latter isomorphic to semidirect product automorphisms H objects. Dropping braiding, we normal basis. For Sweedler cohomology second Picard invertible H-comodules H...

We find a new braided Hopf structure for the algebra satisfied by the entries of the braided matrix BSLq(2). A new nonbraided algebra whose coalgebra structure is the same as the braided one is found to be a two parameter deformed algebra. It is found that this algebra is not a comodule algebra under adjoint coaction. However, it is shown that for a certain value of one of the deformation param...

We show that the differential complex Ω B over the braided matrix algebra BM q (N) represents a covariant comodule with respect to the coaction of the Hopf algebra Ω A which is a differential extension of GL q (N). On the other hand, the algebra Ω A is a covariant braided comodule with respect to the coaction of the braided Hopf algebra Ω B. Geometrical aspects of these results are discussed.

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring C. This allows to extend (weak and...

Let H be a finite dimensional quasi-Hopf algebra over a field k and A a right H-comodule algebra in the sense of [12]. We first show that on the k-vector space A⊗H∗ we can define an algebra structure, denoted by A # H∗, in the monoidal category of left H-modules (i.e. A # H∗ is an Hmodule algebra in the sense of [2]). Then we will prove that the category of two-sided (A,H)bimodules HM H A is is...