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Difference between revisions of "Cartan subgroup"

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<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  C. Chevalley,  "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  C. Chevalley,  "Théorie des groupes de Lie" , '''2–3''' , Hermann  (1951–1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Borel,  "Linear algebraic groups" , Benjamin  (1969)</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</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)</TD></TR></table>
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<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>
  
  
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<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</TD></TR></table>
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<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>

Revision as of 10:02, 24 March 2012

of a group

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

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

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

Let be a connected real Lie group with Lie algebra . Then the Cartan subgroups of are closed in (but not necessarily connected) and their Lie algebras are Cartan subalgebras of . If is an analytic subgroup in and is the smallest algebraic subgroup of containing , then the Cartan subgroups of are intersections of with the Cartan subgroups of . In the case when is compact, the Cartan subgroups are connected, Abelian (being maximal tori) and conjugate to one another, and every element of 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=11216
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article