A Torsion Theory in the Category of Cocommutative Hopf Algebras

Abstract

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.

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.…”

A Seifert–van Kampen theorem in non-abelian algebra

“…This easily provides us with many examples such as groups, (nonunitary) rings, associative algebras, Lie algebras, crossed modules, etc. Further examples of a different kind include loops [7], Heyting semilattices [52], cocommutative Hopf algebras over a field of characteristic zero [38], and the dual of the category of pointed sets [13].…”

“…On the other hand, not every semi-abelian category is: for instance, the categories of loops and of Jordan algebras are not. Examples of algebraically coherent semi-abelian categories are the categories of groups, associative algebras, Lie algebras, Leibniz algebras, Poisson algebras over a commutative ring with unit, all Neumann varieties of groups [59], all Orzech categories of interest [60], next to the categories of rings, crossed modules, and cocommutative Hopf algebras over a field of characteristic zero [38].…”

A semi-abelian extension of a theorem by Takeuchi

“…More recently it was observed [22] that, when the base field K has characteristic zero, the category Hopf K,coc is semi-abelian (in the sense of [28]). This fact implies that many of the exactness properties of the category of cocommutative Hopf algebras follow directly from the axioms of semi-abelian category, and this has opened the way to explore some new connections between categorical algebra and Hopf algebra theory [23,39,20].…”