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Cartan subgroup

## Summary

In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group ${\displaystyle G}$ over a (not necessarily algebraically closed) field ${\displaystyle k}$ is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If ${\displaystyle k}$ is algebraically closed, they are all conjugate to each other. [1]

Notice that in the context of algebraic groups a torus is an algebraic group ${\displaystyle T}$ such that the base extension ${\displaystyle T_{({\bar {k}})}}$ (where ${\displaystyle {\bar {k}}}$ is the algebraic closure of ${\displaystyle k}$) is isomorphic to the product of a finite number of copies of the ${\displaystyle \mathbf {G} _{m}=\mathbf {GL} _{1}}$. 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.

If ${\displaystyle G}$ is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser [2] and thus Cartan subgroups of ${\displaystyle G}$ are precisely the maximal tori.

## Example

The general linear groups ${\displaystyle \mathbf {GL} _{n}}$  are reductive. The diagonal subgroup is clearly a torus (indeed a split torus, since it is product of n copies of ${\displaystyle \mathbf {G} _{m}}$  already before any base extension), and it can be shown to be maximal. Since ${\displaystyle \mathbf {GL} _{n}}$  is reductive, the diagonal subgroup is a Cartan subgroup.