Cartier Structure and Unipotent Radicals of Cartan-Type Automorphism Groups

Let k be a perfect field of characteristic p >= 5. For the height-one special algebra S(n;1)^(1), n >= 3, the reduced identity component of the automorphism scheme is U_S ⋊ GL_n. The congruence filtration of U_S determines its lower central and derived series and, over F_q, its Frattini quotient and lower p-central series. A single Cartier quotient occurs both in the abelianization and in a summand of the five-term homology sequence. The full non-smooth automorphism scheme contains a canonical Frobenius thickening of U_S; its Verschiebung filtration and distribution algebra are computed. At arbitrary divided-power height, Cartier–Verschiebung morphisms on the de Rham complexes induce explicit maps on the corresponding restricted derivations. In height one, the reduced unipotent radicals are determined for the Witt family in rank at least two and for the Hamiltonian family. For the contact family, the full automorphism group scheme is identified with the coordinate contact stabilizer, including the exceptional derived case. For every r >= 2, the Hamiltonian socle line has trivial reduced intersection with the algebraic derived subgroup, while its F_q-points inject into the finite-group abelianization for every q = p^f, including the resonant cases r + 1 ≡ 0 (mod p).