Hopf monoids in varieties
Commutative varieties provide a natural setting for generalizing Hopf algebra theory over commutative rings, since they satisfy the various conditions identified in the category theoretical analysis of this theory to guarantee for example the existence of all naturally occurring forgetful functors in this context, the existence of universal measuring comonoids and the existence of generalized finite duals. It will be shown in addition, that crucial properties of the latter, known from the case of Hopf algebra theory over commutative rings, can be generalized Hopf algebra theory over a commutative variety. The attempt to generalize its construction leads to a couple of questions concerning properties of the monoidal structure of a commutative variety, which seem to be of a more general interest.
KeywordsCommutative varieties Hopf monoids Free and cofree Hopf monoids Universal measuring comonoids Group algebras Finite dual
Mathematics Subject Classification08B99 16T05
I am grateful to the anonymous referee for his of her constructive criticism, which lead to a considerable improvement of the presentation.
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