Self-duality of Hopf algebras

1.1. As an \(R\)-module, \(A = \Lambda_R[\alpha]\) has basis \(\{1, \alpha\}\) with \(\alpha^2 = 0\).The comultiplication is determined by the primitive generator:\[ \Delta(1) = 1 \otimes 1, \qquad \Delta(\alpha) = \alpha \otimes 1 + 1 \otimes \alpha. \]

1.2. Let \(\{1^*, \alpha^*\}\) be the dual basis in \(A^* = \operatorname{Hom}_R(A, R)\).The product in \(A^*\) is given by the dual of the coproduct \(m^* = \Delta^*\):\[ \langle \alpha^* \cdot \alpha^*, \alpha \rangle = \langle \alpha^* \otimes \alpha^*, \Delta(\alpha) \rangle = \langle \alpha^* \otimes \alpha^*, \alpha \otimes 1 + 1 \otimes \alpha \rangle = 0. \]Since \(A^*\) is trivial in degrees \(> \deg(\alpha)\), \((\alpha^*)^2 = 0\), so \(A^* \cong \Lambda_R[\alpha^*]\) as algebras.

1.3. The coproduct on \(A^*\) is given by the dual of multiplication \(\Delta^* = m^*\):\[ \langle \Delta^*(\alpha^*), x \otimes y \rangle = \langle \alpha^*, x y \rangle \implies \Delta^*(\alpha^*) = \alpha^* \otimes 1^* + 1^* \otimes \alpha^*. \]Thus \(\alpha \mapsto \alpha^*\) defines an isomorphism of graded Hopf algebras \(\Lambda_R[\alpha] \cong (\Lambda_R[\alpha])^*\).

2.1. As an \(\mathbb{F}_p\)-vector space, \(A\) has basis \(\{1, \alpha, \alpha^2, \dots, \alpha^{p-1}\}\) with \(\alpha^p = 0\).Since \(\alpha\) is primitive (\(\Delta(\alpha) = \alpha \otimes 1 + 1 \otimes \alpha\)), the coproduct on basis elements is:\[ \Delta(\alpha^k) = (\alpha \otimes 1 + 1 \otimes \alpha)^k = \sum_{j=0}^k \binom{k}{j} \alpha^j \otimes \alpha^{k-j}. \]

2.2. Let \(\{\beta_0, \beta_1, \dots, \beta_{p-1}\}\) be the dual basis of \(A^*\) with \(\langle \beta_i, \alpha^j \rangle = \delta_{ij}\).The product in \(A^*\) satisfies:\[ \langle \beta_i \beta_j, \alpha^k \rangle = \langle \beta_i \otimes \beta_j, \Delta(\alpha^k) \rangle = \binom{k}{i} \delta_{i+j, k} \implies \beta_i \beta_j = \binom{i+j}{i} \beta_{i+j} \quad (i+j < p), \]and \(\beta_i \beta_j = 0\) for \(i+j \ge p\).

2.3. In particular, \(\beta_1^k = k! \beta_k\) for all \(0 \le k < p\).Since \(k < p\), \(k! \in \mathbb{F}_p^\times\), so \(\beta_k = \frac{\beta_1^k}{k!}\).Furthermore, \(\beta_1^p = p! \beta_p = 0\) in \(\mathbb{F}_p\).Thus \(A^* \cong \mathbb{F}_p[\beta_1]/(\beta_1^p)\) as an algebra generated by \(\beta_1\).

2.4. The dual coproduct on \(\beta_1\) satisfies:\[ \langle \Delta^*(\beta_1), \alpha^i \otimes \alpha^j \rangle = \langle \beta_1, \alpha^i \alpha^j \rangle = \begin{cases} 1 & (i, j) \in \{(1, 0), (0, 1)\}, \\ 0 & \text{otherwise,} \end{cases} \]which gives \(\Delta^*(\beta_1) = \beta_1 \otimes 1^* + 1^* \otimes \beta_1\).Thus \(\beta_1\) is primitive.The map \(\alpha \mapsto \beta_1\) defines an isomorphism of graded Hopf algebras \(A \cong A^*\).