Automorphisms of Twisted Chevalley Groups over Commutative Rings

Abstract

We classify all abstract automorphisms of the elementary and full twisted Chevalley groups of types ${}^2{A}_{\ell }$ ($\ell \ge 3$), ${}^2{D}_{\ell }$ ($\ell \ge 4$), ${}^2{E}_6$, and ${}^3{D}_4$ over arbitrary commutative rings, for every invariant isogeny type. The coefficient automorphism $\theta$ has order dividing that of the diagram automorphism. We assume that $2$ is invertible and, for triality, that $3$ is invertible as well. Three features distinguish the answer from the usual inner--diagonal and coefficient decomposition. The elementary groups of types ${}^2{A}_3$ and ${}^2{A}_4$ admit an additive family of exceptional automorphisms supported at $\theta$-fixed maximal ideals with residue field ${\mathbb{F}}_3$. In type ${}^2{D}_4$, diagram automorphisms may vary on idempotent components, subject to compatibility with the involution and the isogeny lattice. In triality, the coefficient automorphisms preserve the trace operator $r\mapsto r+\theta (r)+{\theta }^2(r)$ but need not commute with $\theta$. We determine which elementary automorphisms extend to the full groups and describe the remaining central factors. A nonzero exceptional unitary factor extends precisely in the simply connected type ${}^2{A}_3$, giving a classification for the hyperbolic group ${\mathrm{SU}}_4(R,\theta ;Q_4)$. The proof glues uniquely determined normalization data for a finite root frame and then recovers the coefficient action from root-subgroup relations.

arXiv comment66 pages