# 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

This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article