External Homogenization for Hopf Algebras: Applications to Maschke's Theorem

We apply to Hopf algebras a construction from graded rings, called the group ring of a graded ring. By using this construction we study the transfer of properties between certain categories of relative Hopf modules. As another application, we obtain a Maschke-type theorem for a Galois extension over a semisimple Hopf algebra.     

Quasi-elementary <Emphasis Type="Italic">H</Emphasis>-Azumaya Algebras Arising from Generalized (Anti) Yetter-Drinfeld Modules

Let H be a Hopf algebra with bijective antipode, α, β ∈ Aut _Hopf (H) and M a finite dimensional (α, β)-Yetter-Drinfeld module. We prove that End(M) endowed with certain structures becomes an H-Azumaya algebra, and the set of H-Azumaya algebras of this type is a subgroup of BQ(k, H), the Brauer group of H.     

Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category

If H is a Hopf algebra with bijective antipode and α, β ∈ Aut _Hopf (H), we introduce a category , generalizing both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. We construct a braided T-category having all the categories as components, which, if H is finite dimensional, coincides with the representations of a certain quasitriangular T-coalgebra DT(H) that we construct. We also prove that if (α, β) admits a so-called pair in involution, then is isomorphic to the category of usual Yetter-Drinfeld modules . Research partially supported by the programme CERES of the Romanian Ministry of Education and Research, contract no. 4-147/2004.     

A structure theorem for quasi-Hopf comodule algebras

If is a quasi-Hopf algebra and is a right -comodule algebra such that there exists a morphism of right -comodule algebras, we prove that there exists a left -module algebra such that . The main difference when comparing to the Hopf case is that, from the multiplication of , which is associative, we have to obtain the multiplication of , which in general is not; for this we use a canonical projection arising from the fact that becomes a quasi-Hopf -bimodule.

     

Equivalent Crossed Products and Cross Product Bialgebras

In a previous article we proved a result of the type “invariance under twisting” for Brzeziński's crossed products. In this article we prove a converse of this result, obtaining thus a characterization of what we call equivalent crossed products. As an application, we characterize cross product bialgebras (in the sense of Bespalov and Drabant) that are equivalent (in a certain sense) to a given cross product bialgebra in which one of the factors is a bialgebra and whose coalgebra structure is a tensor product coalgebra.     

Exploring Unknown Undirected Graphs

A robot has to construct a complete map of an unknown environment modeled as an undirected connected graph. The task is to explore all nodes and edges of the graph using the minimum number of edge traversals. The penalty of an exploration algorithm running on a graph G = (V(G), E(G)) is the worst-case number of traversals in excess of the lower bound |E(G)| that it must perform in order to explore the entire graph. We give an exploration algorithm whose penalty is O(|V(G)|) for every graph. We also show that some natural exploration algorithms cannot achieve this efficiency.     

On Quasi-Hopf Smash Products and Twisted Tensor Products of Quasialgebras

We analyze some relations between quasi-Hopf smash products and certain twisted tensor products of quasialgebras. Along the way we obtain also some results of independent interest, such as a duality theorem for finite dimensional quasi-Hopf algebras and a universal property for generalized diagonal crossed products.     

Some Bialgebroids Constructed by Kadison and Connes–Moscovici are Isomorphic

We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.