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.

Notice that in the context of algebraic groups a ''torus'' is an algebraic group $T$
such that the base extension $T_$ (where $\bar$ is the algebraic closure of $k$) is isomorphic to the product of a finite number of copies of the $\mathbf_m=\mathbf_1$. 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 and thus Cartan subgroups of $G$ are precisely the maximal tori.

Example

The general linear groups $\mathbf_n$ are reductive. The diagonal subgroup is clearly a torus (indeed a ''split'' torus, since it is product of n copies of $\mathbf_m$ already before any base extension), and it can be shown to be maximal. Since $\mathbf_n$ is reductive, the diagonal subgroup is a Cartan subgroup.

See also

* Borel subgroup
* Algebraic group
* Algebraic torus

References

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Algebraic geometry
Linear algebraic groups

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