Small Bialgebras with a Projection: Applications
A. Ardizzoni, C. Menini

TL;DR
This paper investigates small bialgebras with projections, focusing on their structure as deformed bosonizations, and explores their properties and examples, especially when the subbialgebra is cosemisimple.
Contribution
It describes the behavior of cocycles in deformed bosonizations of finite-dimensional, thin pre-bialgebras and analyzes their arithmetic properties in the context of cosemisimple subbialgebras.
Findings
Behavior of cocycles in finite-dimensional, thin pre-bialgebras
Analysis of arithmetic properties when H is cosemisimple
Construction of examples with atypical multiplication or non-trivial cocycles
Abstract
In this paper we continue the investigation started in [A.M.St.-Small], dealing with bialgebras with an -bilinear coalgebra projection over an arbitrary subbialgebra with antipode. These bialgebras can be described as deformed bosonizations R#_{\xi} H of a pre-bialgebra by with a cocycle . Here we describe the behavior of in the case when 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 . Meaningful results are obtained when is cosemisimple. By means of Ore extension construction, we provide some examples of atypical situations (e.g. the multiplication of is not -colinear or is non-trivial).
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