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Let $H_1,\cdots, H_n$ be Hopf algebras over the field $\mathbb{Z}_p$, $p$ prime. Then the tensor product $$ \bigotimes_{i=1}^nH_i $$ is still an algebra over $\mathbb{Z}_p$. Is $ \otimes_{i=1}^nH_i $ still a Hopf algebra over $\mathbb{Z}_p$? How to obtain its product, unit, coproduct, augmentation, antipode from the products, units, coproducts, augmentations, antipodes of $H_1,\cdots,H_n$?

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    $\begingroup$ The tensor product of finitely many (I suspect infinitely many, too, but I have not checked) Hopf algebras over a commutative ring is still a Hopf algebra, and its structure maps are obtained as you would you expect: E.g., if $C$ and $D$ are two coalgebras, then $C \otimes D$ becomes a coalgebra with comultiplication $\Delta_{C\otimes D} = \left(\operatorname{id}_C \otimes T \otimes \operatorname{id}_D\right) \circ \left(\Delta_C \otimes \Delta_D\right)$ where $T$ is the twist map $C \otimes D \to D \otimes C$. I'll let you guess the counity. $\endgroup$ – darij grinberg Jul 28 '15 at 13:08
  • $\begingroup$ Dear @darijgrinberg, thanks! could you give some exact references? $\endgroup$ – Shiquan Jul 29 '15 at 3:01
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    $\begingroup$ Most texts on coalgebras define at least the tensor product of two coalgebras. For instance, §1.3 of web.mit.edu/~darij/www/algebra/HopfComb-sols.pdf (though without proof), or §3.0 in Sweedler's "Hopf algebras". Let me know if you want a detailed proof (though I find it rather straightforward). $\endgroup$ – darij grinberg Jul 29 '15 at 3:11
  • $\begingroup$ The link I gave above should be replaced by cip.ifi.lmu.de/~grinberg/algebra/HopfComb-sols.pdf . $\endgroup$ – darij grinberg May 15 at 6:11

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