Classify algebraically the Hopf algebras \(A\) over \(\mathbb{Z}\) such that \(A^n\) is free for each \(n\) and \(A \otimes \mathbb{Q} \approx \mathbb{Q}[\alpha]\). In particular, determine which Hopf algebras \(A \otimes \mathbb{Z}_p\) arise from such \(A\)’s.
Solution
We classify these Hopf algebras as integral Hopf lattices in a rational polynomial Hopf algebra.
Let \(A\) satisfy the hypotheses and let \(d>0\) be the least positive degree in which \(A\otimes\mathbb Q\) is nonzero. Since \[ A\otimes\mathbb Q\cong\mathbb Q[t] \] as a graded algebra, \(|t|=d\) is even. In degree \(d\) there are no positive-degree decomposable terms, so the Hopf axioms force \[ \Delta(t)=t\otimes1+1\otimes t. \] Consequently \[ \Delta(t^n)=\sum_{i=0}^n\binom ni t^i\otimes t^{n-i}. \]
Since each \(A^{nd}\) is a free rank-one subgroup of \(\mathbb Q t^n\), choose generators \[ x_n=q_nt^n, \qquad q_n\in\mathbb Q_{>0}, \qquad q_0=q_1=1, \] where rescaling \(t\) has been used to normalize \(q_1=1\). Then \[ x_ix_j=m_{ij}x_{i+j}, \qquad m_{ij}=\frac{q_iq_j}{q_{i+j}}, \] and \[ \Delta(x_{i+j}) =\sum_{i+j=n}c_{ij}x_i\otimes x_j, \qquad c_{ij}=\binom{i+j}{i}\frac{q_{i+j}}{q_iq_j}. \] Thus \(A\) is an integral Hopf algebra exactly when, for every \(i,j\ge0\), \[ \boxed{m_{ij}\in\mathbb Z_{>0},\qquad c_{ij}=\frac{\binom{i+j}{i}}{m_{ij}}\in\mathbb Z_{>0}.} \] Equivalently, \[ \boxed{m_{ij}\mid\binom{i+j}{i}.} \] Associativity is exactly the cocycle relation \[ \boxed{m_{ij}m_{i+j,k}=m_{jk}m_{i,j+k}} \] with \(m_{i0}=m_{0i}=1\). Conversely, any positive-integer system \((m_{ij})\) satisfying these divisor and cocycle conditions determines \(q_n\) recursively by \[ q_0=q_1=1, \qquad q_{n+1}=\frac{q_n}{m_{n,1}}, \] and the cocycle identity then gives \[ m_{ij}=q_iq_j/q_{i+j} \] for all \(i,j\). The formulas above define a Hopf-stable lattice \[ A=\bigoplus_{n\ge0}\mathbb Z\,q_nt^n\subset\mathbb Q[t]. \] This proves the classification. The two extreme cases are \[ m_{ij}=1, \] which gives the polynomial Hopf algebra \(\mathbb Z[t]\), and \[ m_{ij}=\binom{i+j}{i}, \] which gives the divided-power Hopf algebra \(\Gamma_{\mathbb Z}[t]\).
Now fix a prime \(p\) and reduce modulo \(p\). With \(\bar x_n\) denoting the image of \(x_n\), every reduction arising from an integral form above has \[ \bar x_i\bar x_j=\overline{m_{ij}}\,\bar x_{i+j}, \] \[ \Delta(\bar x_n) =\sum_{i=0}^n \overline{\frac{\binom ni}{m_{i,n-i}}} \,\bar x_i\otimes\bar x_{n-i}. \] Hence the mod-\(p\) Hopf algebra is determined by the \(p\)-adic valuations \[ e_{ij}=v_p(m_{ij}). \] They satisfy \[ 0\le e_{ij}\le v_p\binom{i+j}{i}, \] and the additive cocycle relation \[ e_{ij}+e_{i+j,k}=e_{jk}+e_{i,j+k}. \] Conversely, any such \(p\)-adic divisor cocycle lifts: take \[ m_{ij}=p^{e_{ij}} \] (and choose valuation \(0\) at all other primes), then the preceding integral construction produces a Hopf algebra whose reduction has exactly this structure. Up to rescaling each one-dimensional graded piece by a unit of \(\mathbb F_p\), these are precisely the Hopf algebras \(A\otimes\mathbb F_p\) that arise.
In particular, the polynomial and divided-power reductions are the two extreme valuation choices \(e_{ij}=0\) and \[ e_{ij}=v_p\binom{i+j}{i}, \] while intermediate divisor cocycles give the mixed integral forms.