The purpose of this article is to prove that the category of cocommutative Hopf K-algebras, over a field K of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie K-algebras, respectively.

We describe the split extension classifiers in the semi-abelian category of cocommutative Hopf algebras over an algebraically closed field of characteristic zero. The categorical notions of centralizer and of center in the category of cocommutative Hopf algebras is then explored. We show that the categorical notion of center coincides with the one that is considered in the theory of general Hopf algebras.3.4.1, that uses an explicit description of split exact sequences in Hopf K,coc in terms of semi-direct products (also called smash products) of cocommutative Hopf algebras. We then analyse the abstract notions of center and of centralizer, introduced by D. Bourn and G. Janelidze in [8], in the category Hopf K,coc . This part is also based on some nice results concerning centralizers due to A. Cigoli and S. Mantovani in [11]. We finally compare our description of the center in Hopf K,coc with the defintion given by A. Chirvasitu and P. Kasprzak in [10] for arbitrary (not necessarily cocommutative) Hopf algebras, and observe that they coincide in the cocommutative case. Since our results heavily rely on the Cartier-Gabriel-Konstant-Milnor-Moore decomposition theorem for cocommutative Hopf algebras which is only valid over an algebraically closed field of characteristic zero, our description only holds in this case. It remains an open question if this description can be extended to the general case. Acknowledgement. The authors are grateful to Tim Van der Linden for an important remark on a preliminary version of this paper, and to the referee for some useful suggestions.

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