View source for 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|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|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====
<table><tr><td valign="top">[1a]</td> <td valign="top">  C. Chevalley,   "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)  {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}} </td></tr><tr><td valign="top">[1b]</td> <td valign="top">  C. Chevalley,   "Théorie des groupes de Lie" , '''2–3''' , Hermann  (1951–1955)  {{MR|0068552}} {{MR|0051242}} {{MR|0019623}} {{ZBL|0186.33104}} {{ZBL|0054.01303}} {{ZBL|0063.00843}} </td></tr><tr><td valign="top">[2]</td> <td valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </td></tr><tr><td valign="top">[3]</td> <td valign="top">  A. Borel,   J. Tits,   "Groupes réductifs"  ''Publ. Math. IHES'' , '''27'''  (1965)  pp. 55–150  {{MR|0207712}} {{ZBL|0145.17402}} </td></tr><tr><td valign="top">[4]</td> <td valign="top">  M. Demazure,   A. Grothendieck,   "Schémas en groupes I-III" , ''Lect. notes in math.'' , '''151–153''' , Springer  (1970)    {{MR|}} {{ZBL|}} </td></tr></table>



====Comments====


====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  A. Borel,   T.A. Springer,   "Rationality properties of linear algebraic groups"  ''Tohoku Math. J. (2)'' , '''20'''  (1968)  pp. 443–497  {{MR|0244259}} {{ZBL|0211.53302}} </td></tr></table>