Tensor product of Hopf algebras

Let \(A\) and \(B\) be graded Hopf algebras over the same commutative coefficient ring. Give \(A\otimes B\) the usual graded tensor-product algebra structure \[ (a\otimes b)(a'\otimes b') =(-1)^{|b||a'|}aa'\otimes bb'. \]

Write the coproducts in Sweedler notation as \[ \Delta_A(a)=\sum a_{(1)}\otimes a_{(2)}, \qquad \Delta_B(b)=\sum b_{(1)}\otimes b_{(2)}. \] Define \[ \Delta_{A\otimes B}(a\otimes b) = \sum (-1)^{|a_{(2)}||b_{(1)}|} (a_{(1)}\otimes b_{(1)}) \otimes (a_{(2)}\otimes b_{(2)}). \] Equivalently, this is the composite \[ A\otimes B \xrightarrow{\Delta_A\otimes\Delta_B} (A\otimes A)\otimes(B\otimes B) \longrightarrow (A\otimes B)\otimes(A\otimes B), \] where the second map interchanges the middle two factors with the Koszul sign.

Coassociativity follows immediately from coassociativity of \(\Delta_A\) and \(\Delta_B\) together with associativity of the graded symmetry. The counit is \[ \varepsilon_A\otimes\varepsilon_B, \] and the two counit identities follow factorwise. Finally, \(\Delta_{A\otimes B}\) is an algebra homomorphism because \(\Delta_A\) and \(\Delta_B\) are algebra homomorphisms and the Koszul sign in the middle interchange is exactly the sign required by the graded tensor-product multiplication.

Thus \(A\otimes B\) satisfies the Hopf-algebra axioms.