Cartan subgroup explained

over a (not necessarily algebraically closed) field

is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If

is algebraically closed, they are all conjugate to each other.

Notice that in the context of algebraic groups a torus is an algebraic group

such that the base extension

T_(\bar{k)}

(where

\bar{k}

is the algebraic closure of

) is isomorphic to the product of a finite number of copies of the

G_m=GL₁

. Maximal such subgroups have in the theory of algebraic groups a role that is similar to that of maximal tori in the theory of Lie groups.

is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser and thus Cartan subgroups of

are precisely the maximal tori.

Example

The general linear groups

GL_n

are reductive. The diagonal subgroup is clearly a torus (indeed a split torus, since it is product of n copies of

G_m

already before any base extension), and it can be shown to be maximal. Since

GL_n

is reductive, the diagonal subgroup is a Cartan subgroup.