# tensor products of Hopf algebras

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 $H_1,\cdots,H_n$?

• 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. – darij grinberg Jul 28 '15 at 13:08
• Dear @darijgrinberg, thanks! could you give some exact references? – Shiquan Jul 29 '15 at 3:01
• 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). – darij grinberg Jul 29 '15 at 3:11
• The link I gave above should be replaced by cip.ifi.lmu.de/~grinberg/algebra/HopfComb-sols.pdf . – darij grinberg May 15 at 6:11