Cartan subgroup

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

Notice that in the context of algebraic groups a torus is an algebraic group $T$ such that the base extension $T_{(\overline{k})}$ (where $\overline{k}$ is the algebraic closure of $k$) is isomorphic to the product of a finite number of copies of the ${𝐆}_m={𝐆𝐋}_1$. 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 $G$ is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser Template:Sfnp and thus Cartan subgroups of $G$ 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.

See also

• Borel subgroup
• Algebraic group
• Algebraic torus

References

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