Different Hopf algebra structures on same graded algebra 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?
 A: To clarify the Theorem I would say, ".... If A is a graded commutative Hopf algebra then, the underlying algebra structure is tensor of exterior and a polynomial."
And on the same underlying algebra, one can have two different coproduct
Let $Q[x]$ be polynomial of algebra where $|x| =0$. A coproduct will be a map of $Q[x]$-algebra, and its enough to define the map on $|x|$. So I will describe them as follows 
$\psi_{1}(x)= x \otimes 1 + 1 \otimes x$ 
$\psi_{2}(x)= x \otimes x$.
A: 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.
