In this paper we continue the investigation started in \cite{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).
Small Bialgebras with a Projection: Applications
ARDIZZONI, Alessandro;MENINI, Claudia
2009
Abstract
In this paper we continue the investigation started in \cite{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).I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
