View source for Rank of a Lie group

''(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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====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>