Small Bialgebras with a Projection: Applications

In this paper we continue the investigation started in [AMStu], dealing with bialgebras A with an H-bilinear coalgebra projection over an arbitrary subbialgebra H with antipode. These bialgebras can be described as deformed bosonizations R#ξH of a pre-bialgebra R by H with a cocycle ξ. Here we describe the behavior of ξ in the case when R is f.d. and thin i.e. it is connected with one dimensional space of primitive elements. This is used to analyze the arithmetic properties of A. Meaningful results are obtained when H is cosemisimple. By means of Ore extension construction, we provide some examples of atypical situations (e.g. the multiplication of R is not H-colinear or ξ is non-trivial).