Small Bialgebras with a Projection: Applications

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 $A$ with an $H$-bilinear coalgebra projection over an arbitrary subbialgebra $H$ with antipode. These bialgebras can be described as deformed bosonizations $R#_{\xi} H$ of a pre-bialgebra $R$ by $H$ with a cocycle $\xi$. Here we describe the behavior of $\xi$ 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 $\xi$ is non-trivial).