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Difference between revisions of "Rank of a Lie group"

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''(real or complex)''
 
''(real or complex)''
  
The (real, respectively, complex) dimension of any [[Cartan subgroup|Cartan subgroup]] of it. The rank of a Lie group coincides with the rank of its Lie algebra (see [[Rank of a Lie algebra|Rank of a Lie algebra]]). If a Lie group $G$ coincides with the set of real or complex points of a [[Linear algebraic group|linear algebraic group]] $\widehat G$, then the rank of $G$ coincides with the rank of $\widehat G$ (cf. [[Rank of an algebraic group|Rank of an algebraic group]]).
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The (real, respectively, complex) dimension of any [[Cartan subgroup]] of it. The rank of a Lie group coincides with the rank of its Lie algebra (see [[Rank of a Lie algebra]]). If a Lie group $G$ coincides with the set of real or complex points of a [[linear algebraic group]] $\widehat G$, then the rank of $G$ coincides with the rank of $\widehat G$ (cf. [[Rank of an algebraic group]]).
 
 
 
 
 
 
====Comments====
 
 
 
  
 
====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) pp. Chapt. 1</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) Chapt. 1 {{ISBN|0-471-13735-9}}</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top"> V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) {{MR|0376938}} {{ZBL|0371.22001}} </TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top"> V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) {{MR|0376938}} {{ZBL|0371.22001}} </TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top"> A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) {{MR|0855239}} {{ZBL|0604.22001}} </TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top"> A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) {{MR|0855239}} {{ZBL|0604.22001}} </TD></TR>

Latest revision as of 10:43, 19 May 2024

(real or complex)

The (real, respectively, complex) dimension of any Cartan subgroup of it. The rank of a Lie group coincides with the rank of its Lie algebra (see Rank of a Lie algebra). If a Lie group $G$ coincides with the set of real or complex points of a linear algebraic group $\widehat G$, then the rank of $G$ coincides with the rank of $\widehat G$ (cf. Rank of an algebraic group).

References

[a1] R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) Chapt. 1 ISBN 0-471-13735-9
[a2] V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) MR0376938 Zbl 0371.22001
[a3] A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) MR0855239 Zbl 0604.22001
[a4] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 MR0396773 Zbl 0325.20039
How to Cite This Entry:
Rank of a Lie group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_of_a_Lie_group&oldid=34499
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article