A semi-abelian extension of a theorem by Takeuchi

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:223

Article

Gran, Marino¹  Sterck, Florence¹  Vercruysse, Joost² 

[1] Catholic Univ Louvain, Inst Rech Math & Phys, Chemin Cyclotron 2, B-1348 Louvain La Neuve, Belgium

[2] Univ Libre Bruxelles, Dept Math, Brussels, Belgium

关键词: Cocommutative Hopf algebras;    Semi-abelian categories;    Crossed modules;    Internal groupoids;    Categorical commutator;   

DOI  :  10.1016/j.jpaa.2019.01.004

来源: Elsevier

【 摘 要 】

We prove that the category of cocommutative Hopf algebras over a field is a semiabelian 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. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free