I'm reading Hatcher's book on algebraic topology. In Section 3.C, he proves as Theorem 3C.4 that if $A$ is a graded commutative associative Hopf algebra over a field of characteristic $0$ and $A^n$ is finitely generated for each $n$, then $A$ is isomorphic **as an algebra** to the tensor product of an exterior algebra on odd-dimensional generators and a polynomial algebra on even-dimensional generators. I emphasized the phrase **as an algebra** because I first assumed that this was a typo on his part, but indeed in his proof he only shows that $A$ has the right algebra structure. The coproduct is used to establish this, but he does not prove that $A$ has the same coproduct as the indicated tensor product.


Question : Must it? In other words, over a field of characteristic $0$ can the tensor product of an exterior algebra on odd-dimensional generators and a polynomial algebra on even-dimensional generators be given a Hopf-algebra structure other than the one coming from the tensor product?



Different Hopf algebra structures on same graded algebra
There are generalizations of the following construction, but here is a simple one. Let $\mathfrak n$ denote the Lie algebra of strictly-upper-triangular $n\times n$ matrices. It is $\mathbb Z\_{>0}$-graded, by declaring that for $1\leq k \leq n$, the degree-$k$ part of $\mathfrak n$ consists of those matrices supported on the diagonal that is $k$ steps above the main diagonal. (So $\mathfrak n\_k$ has a basis consisting of those matrices with a $1$ in the $(i,i+k)$th spot and $0$s elsewhere.) This is a bosonic grading, so to match Hatcher's conventions, you perhaps should double the grading.


In any case, consider the polynomial algebra $\operatorname{Sym}(\mathfrak n^\ast)$ generated by the dual vector space — so this is the algebra $\mathcal{O}(\mathfrak n)$ of polynomial functions on $\mathfrak n$. It is connected and $\mathbb Z\_{\geq 0}$-graded. It has many compatible coproducts. One is the canonical coproduct generated by $x \mapsto x\otimes 1 + 1\otimes x$ for $x\in \mathfrak n^\ast$. Another is the Baker–Campbell–Hausdorff formula $\operatorname{BCH}$. In detail, let $N$ denote the group of matrices which have $1$s on the diagonal and $0$s below it, and let $\operatorname{m} : N\times N \to N$ denote the map of matrix multiplication. Then the matrix $\exp : \mathfrak{n} \to N$ is a degree-$n$ polynomial, as is its inverse $\log: N \to \mathfrak{n}$; we can therefore pull back polynomial functions along these, so for example we have $\operatorname{m}^\ast : \mathcal{O}(N) \to \mathcal{O}(N)\otimes \mathcal{O}(N)$. Then:
$$\operatorname{BCH} = (\exp^\ast \otimes \exp^\ast) \circ \operatorname{m}^\ast \circ \log^\ast.$$
It is a standard fact that $\operatorname{BCH}$ preserves the grading on $\operatorname{Sym}(\mathfrak n^\ast)$, and makes $\operatorname{Sym}(\mathfrak n^\ast)$ into a Hopf algebra.


