# Cartan subgroup

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of a group $G$

A maximal nilpotent subgroup $C$ of $G$ each normal subgroup of finite index of which has finite index in its normalizer in $G$ . If $G$ is a connected linear algebraic group over a field of characteristic zero, then a Cartan subgroup of $G$ can also be defined as a closed connected subgroup whose Lie algebra is a Cartan subalgebra of the Lie algebra of $G$ . An example of a Cartan subgroup is the subgroup $D$ of all diagonal matrices in the group $\mathop{\rm GL}\nolimits _{n} (k)$ of all non-singular matrices.

In a connected linear algebraic group $G$ , a Cartan subgroup can also be defined as the centralizer of a maximal torus of $G$ , or as a connected closed nilpotent subgroup $C$ which coincides with the connected component of the identity (the identity component) of its normalizer in $G$ . The sets $C _{s}$ and $C _{u}$ of all semi-simple and unipotent elements of $C$ ( see Jordan decomposition) are closed subgroups in $C$ , and $C = C _{s} \times C _{u}$ . In addition, $C _{s}$ is the unique maximal torus of $G$ lying in $C$ . The dimension of a Cartan subgroup of $G$ is called the rank of $G$ . The union of all Cartan subgroups of $G$ contains an open subset of $G$ with respect to the Zariski topology (but is not, in general, the whole of $G$ ). Every semi-simple element of $G$ lies in at least one Cartan subgroup, and every regular element in precisely one Cartan subgroup. If $\phi : \ G \rightarrow G ^ \prime$ is a surjective morphism of linear algebraic groups, then the Cartan subgroups of $G ^ \prime$ are images with respect to $\phi$ of Cartan subgroups of $G$ . Any two Cartan subgroups of $G$ are conjugate. A Cartan subgroup of a connected semi-simple (or, more generally, reductive) group $G$ is a maximal torus in $G$ .

Let the group $G$ be defined over a field $k$ . Then there exists in $G$ a Cartan subgroup which is also defined over $k$ ; in fact, $G$ is generated by its Cartan subgroups defined over $k$ . Two Cartan subgroups of $G$ defined over $k$ need not be conjugate over $k$ ( but in the case when $G$ is a solvable group, they are conjugate). The variety of Cartan subgroups of $G$ is rational over $k$ .

Let $G$ be a connected real Lie group with Lie algebra $\mathfrak g$ . Then the Cartan subgroups of $G$ are closed in $G$ ( but not necessarily connected) and their Lie algebras are Cartan subalgebras of $\mathfrak g$ . If $G$ is an analytic subgroup in $\mathop{\rm GL}\nolimits _{n} ( \mathbf R )$ and $\overline{G}$ is the smallest algebraic subgroup of $\mathop{\rm GL}\nolimits _{n} ( \mathbf R )$ containing $G$ , then the Cartan subgroups of $G$ are intersections of $G$ with the Cartan subgroups of $\overline{G}$ . In the case when $G$ is compact, the Cartan subgroups are connected, Abelian (being maximal tori) and conjugate to one another, and every element of $G$ lies in some Cartan subgroup.

#### References

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