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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077460/r0774601.png" /> coincides with the set of real or complex points of a [[Linear algebraic group|linear algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077460/r0774602.png" />, then the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077460/r0774603.png" /> coincides with the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077460/r0774604.png" /> (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]]).
  
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====References====
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<table>
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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>
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<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>
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<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>
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<TR><TD valign="top">[a4]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR>
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</table>
  
 
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[[Category:Lie theory and generalizations]]
====Comments====
 
 
 
 
 
====References====
 
<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><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.S. Varadarajan,  "Lie groups, Lie algebras, and their representations" , Prentice-Hall  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.W. Knapp,  "Representation theory of semisimple groups" , Princeton Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.E. Humphreys,  "Linear algebraic groups" , Springer  (1975)  pp. Sect. 35.1</TD></TR></table>
 

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=12510
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article