Commutator subgroup of a Lie group - Imported Knowlpedia Content

Let $G$ be a Lie group.

Definition. The commutator subgroup (or derived subgroup) $[G,G]$ is the subgroup generated by all commutators

[g,h]=ghg^{-1}h^{-1}\quad (g,h\in G).

It is the smallest normal subgroup $N\triangleleft G$ such that $G/N$ is abelian; equivalently, $G/[G,G]$ is the abelianization of $G$ .

Remarks

Lie-theoretic relation. If $\mathfrak{g}=\mathrm{Lie}(G)$ , then the Lie algebra of the identity component of $[G,G]$ is the derived subalgebra $[\mathfrak{g},\mathfrak{g}]$ . This makes $[G,G]$ the global counterpart of “taking brackets” in $\mathfrak{g}$ .

Topological subtlety. In general, $[G,G]$ need not be closed. When forming a smooth quotient, one often replaces it by its closure $\overline{[G,G]}$ , which is a closed subgroup and hence a Lie subgroup by the closed subgroup theorem.

Context. $G$ is abelian precisely when $[G,G]$ is trivial, and the size of $[G,G]$ controls how far $G$ is from being commutative.