Automorphisms of Twisted Chevalley Groups over Commutative Rings

@misc{bunina2026automorphisms,
  title = {{Automorphisms of Twisted Chevalley Groups over Commutative Rings}},
  author = {Elena I. Bunina and Deep H. Makadiya},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30314},
  eprint = {2610.30314},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30314},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Automorphisms of Twisted Chevalley Groups over Commutative Rings
AU  - Elena I. Bunina
AU  - Deep H. Makadiya
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30314
UR  - https://arxiv.org/abs/2610.30314
AB  - We classify all abstract automorphisms of the elementary and full twisted Chevalley groups of types ^2A_ℓ (ℓ≥3), ^2D_ℓ (ℓ≥4), ^2E_6, and ^3D_4 over arbitrary commutative rings, for every invariant isogeny type. The coefficient automorphism θ 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 ^2A_3 and ^2A_4 admit an additive family of exceptional automorphisms supported at θ-fixed maximal ideals with residue field 𝔽_3. In type ^2D_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↦ r+θ(r)+θ^2(r) but need not commute with θ. 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 ^2A_3, giving a classification for the hyperbolic group SU_4(R,θ;Q_4). The proof glues uniquely determined normalization data for a finite root frame and then recovers the coefficient action from root-subgroup relations.
ER  -