User:Maximilian Janisch/latexlist/Algebraic Groups/Cartan subgroup

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of a group $k$

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

In a connected linear algebraic group $k$, a Cartan subgroup can also be defined as the centralizer of a maximal torus of $k$, or as a connected closed nilpotent subgroup $C$ which coincides with the connected component of the identity (the identity component) of its normalizer in $k$. The sets $C$ and $C$ 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, $C$ is the unique maximal torus of $k$ lying in $C$. The dimension of a Cartan subgroup of $k$ is called the rank of $k$. The union of all Cartan subgroups of $k$ contains an open subset of $k$ with respect to the Zariski topology (but is not, in general, the whole of $k$). Every semi-simple element of $k$ 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 $( 1 )$ of Cartan subgroups of $k$. Any two Cartan subgroups of $k$ are conjugate. A Cartan subgroup of a connected semi-simple (or, more generally, reductive) group $k$ is a maximal torus in $k$.

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

Let $k$ be a connected real Lie group with Lie algebra $8$. Then the Cartan subgroups of $k$ are closed in $k$ (but not necessarily connected) and their Lie algebras are Cartan subalgebras of $8$. If $k$ is an analytic subgroup in $GL _ { n } ( R )$ and $C$ is the smallest algebraic subgroup of $GL _ { n } ( R )$ containing $k$, then the Cartan subgroups of $k$ are intersections of $k$ with the Cartan subgroups of $C$. In the case when $k$ is compact, the Cartan subgroups are connected, Abelian (being maximal tori) and conjugate to one another, and every element of $k$ lies in some Cartan subgroup.

References

[1a]	C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) MR0082628 MR0015396 Zbl 0063.00842
[1b]	C. Chevalley, "Théorie des groupes de Lie" , 2–3 , Hermann (1951–1955) MR0068552 MR0051242 MR0019623 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843
[2]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[3]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[4]	M. Demazure, A. Grothendieck, "Schémas en groupes I-III" , Lect. notes in math. , 151–153 , Springer (1970)

References

[a1]	A. Borel, T.A. Springer, "Rationality properties of linear algebraic groups" Tohoku Math. J. (2) , 20 (1968) pp. 443–497 MR0244259 Zbl 0211.53302