Maximal toroids and Cartan subgroups of algebraic groups

TL;DR

This paper develops a unified theory of Cartan subgroups and maximal toroids for all affine algebraic groups, establishing their existence, invariance, and correspondence, extending classical results to non-smooth cases.

Contribution

It introduces a comprehensive framework connecting Cartan subgroups and maximal toroids applicable to all affine algebraic groups, including non-smooth ones.

Findings

Maximal toroids always exist in affine algebraic groups.

Maximal toroids are invariant under base change.

There is a natural one-to-one correspondence between maximal toroids and Cartan 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.