Classification des objets galoisiens d'une algèbre de Hopf

Abstract : This thesis is about classification of Galois objects of a Hopf algebra. The notion of Galois extensions, which was intensively studied these last years, is a generalisation of the usual notion of Galois extensions for fields, but also a non commutative analogue of the notion of principal fibre bundles. If $H$ is a Hopf algebra, a $H$-comodule algebra $(Z,\delta : Z\to Z \otimes H)$ is a Galois $H$-extension of a subalgebra $B\subset Z$ if the set of coinvariant elements of $Z$ is equal to $B$ and if the canonical map $\beta : Z \otimes _B Z \to Z\otimes H$ defined by
$$\beta (x\otimes y) = \delta (x) (y\otimes 1)$$ is a bijection. The galois objects are an important class of galois extensions ; these are the one whose coinvariant elements are reduced to the base ring.
Whereas lots of articles deal with galois extensions, we have few classification results up to isomorphism. To get through this problem, Kassel has introduced and developped with Schneider an equivalence relation on galois extension named homotopy.

In this thesis, we give classification results up to homotopy and isomorphism. We deal with the classification of galois object on three axes.
\begin{itemize}
\item[a)] The explicit construction of a galois object in each homotopy class when the Hopf algebra is the quantum group $U_q(\mathfrak{g})$ associated by Drinfeld and Jimbo to a Lie algebra $\mathfrak{g}$, expliciting a theorem of Kassel and Schneider.
\item[b)]The study of galois objects of the quantum group $O_q(SL(2))$ of functions over the group $SL(2)$, and then a classification result in infinite dimension ; we give the classification up to isomorphism and partial results for the classification up to homotopy.
\item[c)] A systematic study of the classification up to isomorphism and homotopy for the Hopf algebras of dimension $\leq 15$ ; we summaryze results dispersed in the litterature concerning families of pointed or semisimple Hopf algebras and add the classification of the galois objects of the Hopf algebra of dimension $8$ which is neither pointed nor semisimple.
\end{itemize}
Document type :
Theses
Mathématiques [math]. Université Louis Pasteur - Strasbourg I, 2007. Français. <NNT : 2007STR13026>
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Submitted on : Monday, June 4, 2007 - 10:54:36 AM
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Thomas Aubriot. Classification des objets galoisiens d'une algèbre de Hopf. Mathématiques [math]. Université Louis Pasteur - Strasbourg I, 2007. Français. <NNT : 2007STR13026>. <tel-00151368>

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