Abstract |
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The aim of this thesis is to establish some new interactions between the theory of semi-abelian categories and the theory of Hopf algebras. We prove that the category of cocommutative Hopf algebras over a field K of characteristic zero is a semi-abelian category, by using the Cartier-Gabriel-Kostant-Milnor-Moore decomposition theorem of any cocommutative Hopf algebra as a semi-direct product of a group Hopf algebra and a primitive Hopf algebra. Then, we show that this category contains a torsion theory whose torsion-free and torsion parts are given, up to equivalence, by the category of groups and by the category of Lie K-algebras, respectively. Then we prove that the category of cocommutative Hopf algebras over an algebraically closed field K of characteristic zero is actually an action representable semi-abelian category, and we describe the split extension classifiers in it in terms of the split extension classifiers in the category of groups and in the category of Lie K-algebras. The categorical notions of centralizer and of center in the category of cocommutative Hopf algebras are then explored, and we show that our description of centers in this category coincides with the ones that have been recently introduced and investigated in the context of general Hopf algebras. |