# Cartan subgroup

*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

[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) |

#### Comments

#### 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 |

**How to Cite This Entry:**

Cartan subgroup.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cartan_subgroup&oldid=44273