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)

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