On Epimorphisms and Monomorphisms of Hopf Algebras
نویسنده
چکیده
We notice that for a Hopf algebra H , its antipode S is both an epimorphism and a monomorphism from H to H in the category of Hopf algebras over a field. Together with the existence of Hopf algebras with non-injective or non-surjective antipode, this proves the existence of non-surjective epimorphisms and non-injective monomorphisms in the category of Hopf algebras. Using Schauenburg’s free Hopf algebra with a bijective antipode on a Hopf algebra and its “dual” universal construction, we then show that one can even find examples of non-surjective (non-injective) epis (resp. monos) of Hopf algebras K → H such that the antipode of H (resp. K) is bijective. Also, we emphasize some connections between the epi/surjective problem and Kaplansky’s 1’st conjecture.
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