Difference between revisions of "Lie group, derived"
From Encyclopedia of Mathematics
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| − | The [[Commutator subgroup|commutator subgroup]] of a [[Lie group|Lie group]]. For any Lie group | + | {{TEX|done}} |
| + | The [[Commutator subgroup|commutator subgroup]] of a [[Lie group|Lie group]]. For any Lie group $G$ its derived Lie group $[G,G]$ is a normal (not necessarily closed) Lie subgroup of $G$. The corresponding ideal of the [[Lie algebra|Lie algebra]] $\mathfrak g$ of the group $G$ coincides with the commutator algebra $[\mathfrak g,\mathfrak g]$ (also called the derived Lie algebra of $\mathfrak g$). The commutator subgroup of a simply-connected (or connected linear) Lie group $G$ is always closed in $G$. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Chevalley, "Theory of Lie groups" , '''1''' , Princeton Univ. Press (1946)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Chevalley, "Theory of Lie groups" , '''1''' , Princeton Univ. Press (1946)</TD></TR></table> | ||
Latest revision as of 16:05, 1 May 2014
The commutator subgroup of a Lie group. For any Lie group $G$ its derived Lie group $[G,G]$ is a normal (not necessarily closed) Lie subgroup of $G$. The corresponding ideal of the Lie algebra $\mathfrak g$ of the group $G$ coincides with the commutator algebra $[\mathfrak g,\mathfrak g]$ (also called the derived Lie algebra of $\mathfrak g$). The commutator subgroup of a simply-connected (or connected linear) Lie group $G$ is always closed in $G$.
References
| [1] | C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) |
How to Cite This Entry:
Lie group, derived. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_derived&oldid=17709
Lie group, derived. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_derived&oldid=17709
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article