A semi-abelian extension of a theorem by Takeuchi

Abstract

We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor-Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context.

27 pages

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Cited by in corpus (14)

• Variations of the Shifting Lemma and Goursat categories
• Torsion theories and coverings of preordered groups
• Facets of congruence distributivity in Goursat categories
• Correspondence theorems for Hopf algebroids with applications to affine groupoids
• An introduction to regular categories
• Non-associative algebras
• Conditional flatness, fiberwise localizations, and admissible reflections
• Algebraic logoi
• The Zassenhaus lemma in star-regular categories
• Split extensions and actions of bialgebras and Hopf algebras
• Action representability of the category of internal groupoids
• Coalgebraic models of Omega-groups
• The Moore Complex of a Simplicial Cocommutative Hopf Algebra
• Braided Hopf Crossed Modules Through Simplicial Structures