Maximal toroids and Cartan subgroups of algebraic groups
Damian Sercombe
TL;DR
The work develops a unified framework for Cartan subgroups and maximal toroids in affine algebraic groups, extending classical Cartan-torus theory beyond smooth groups. It proves the existence of maximal toroids for any affine group, establishes base-change invariance, and reveals a natural bijection between maximal toroids and Cartan subgroups via $C = Z_G(T)^{\circ}$ and $T = Z_G(C)^{\circ}_s$, with further characterization through Frobenius kernels and related corollaries. By connecting to the Demazure-Gabriel correspondence and height-1 groups, the paper generalizes results for restricted Lie algebras and analyzes generation questions, including when a group is or is not generated by Cartan subgroups. Overall, the results provide structural rigidity tools and illuminate generation problems in affine algebraic groups and their Frobenius-layered subgroups.
Abstract
We introduce a unified theory of Cartan subgroups and maximal toroids - defined as connected multiplicative type subgroups that are maximal amongst all such subgroups - which holds for all affine algebraic groups over a field, regardless of smoothness. For instance we show that maximal toroids always exist, that they are invariant under base change, and that they are in natural 1-1 correspondence with Cartan subgroups. Our results generalise known results for Cartan subgroups and maximal tori of smooth affine algebraic groups, as well as their analogues for restricted Lie algebras. We conclude with some applications to, and a brief discussion of, some generation problems for algebraic groups.