Difference between revisions of "Rank of a Lie group"
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| − | 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 | + | 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]]). |
Revision as of 16:22, 19 April 2014
(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).
Comments
References
| [a1] | R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) pp. Chapt. 1 |
| [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=21915
Rank of a Lie group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_of_a_Lie_group&oldid=21915
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article