Difference between revisions of "Rank of a Lie group"
From Encyclopedia of Mathematics
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| − | The (real, respectively, complex) dimension of any [[ | + | 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==== | ====References==== | ||
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| − | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) 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) {{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 19:00, 7 April 2023
(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 |
| [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
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