Split extension classifiers in the category of cocommutative Hopf algebras

Abstract

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.

Some remarks on protolocalizations and protoadditive reflections

“…is defined by ∆(g) = g ⊗ g, for all g in G. Accordingly, for any Hopf K-algebra H, an element h in H is said to be a group-like element if ∆(h) = h ⊗ h for the comultiplication ∆ of H. Identifying the category of cocommutative Hopf K-algebras with the category Grp(CCoAlg K ) of internal groups in the category of cocommutative K-coalgebras (see [29] for instance), and identifying the category of Hopf K-algebras of the form K[G] (for all groups G) with the category of groups, yields now the group-like element functor Grp(CCoAlg K ) → Grp. As shown in [16] (see also [17]), when K is a field of characteristic zero, Grp(CCoAlg K ) is semi-abelian and this functor is a localization.…”

Algebras with representable representations

“…In Grp, the actor of an object X is the automorphism group Aut(X); in Lie K , the actor of X is the Lie algebra of its derivations Der (X). More examples are studied in [4], and new examples continue to be studied, as for crossed modules (in [28]) and for cocommutative Hopf algebras (in [17]). Proposition 1.10.…”

“…For instance, from the output in the ancillary files, it is immediately clear that f 140 can be removed from the system while maintaining its inconsistency. Actually, the system may be reduced significantly: we checked that equations number 1, 3,4,6,7,10,11,12,13,14,15,16,17,18,19,20,22,23,25,26,27,28,29,30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,56,82, 92 together still form an inconsistent system of polynomial equations. However, giving the explicit polynomials that prove the system's inconsistency becomes harder as its size goes down.…”

Equivalences of (co)module algebra structures over Hopf algebras

“…and there exists a unique Hopf algebra homomorphism ϕ 1 : H 1 → R(ζ(H)) such that the following diagram commutes: In the case of an algebraically closed field of characteristic 0, the Cartier-Gabriel-Kostant theorem (see e.g. [27, Corollary 5.6.4 and Theorem 5.6.5]) allows us to give a concrete description of the universal cocommutative Hopf algebra, using a similar technique as in [23].…”