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Difference between revisions of "Lie group, derived"

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The [[Commutator subgroup|commutator subgroup]] of a [[Lie group|Lie group]]. For any Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586201.png" /> its derived Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586202.png" /> is a normal (not necessarily closed) Lie subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586203.png" />. The corresponding ideal of the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586204.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586205.png" /> coincides with the commutator algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586206.png" /> (also called the derived Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586207.png" />). The commutator subgroup of a simply-connected (or connected linear) Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586208.png" /> is always closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586209.png" />.
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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=32088
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article