Suppose that $G$ is a non-Abelian Lie group. Is there a characterization of such groups $G$ for which $$ Z(G)\cong G/[G,G]? $$
(Besides the case where $G$ is itself Abelian)?
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What are some non-abelian examples of this? – Randall Jun 04 '19 at 03:00
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I can't think of any non-trivial ones – AB_IM Jun 04 '19 at 03:13
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1Then I must ask: why do you care? – Randall Jun 04 '19 at 03:27
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1To see if it Z(G) ever becomes functorial (when restricted)...out of curiosity – AB_IM Jun 04 '19 at 03:29
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Here is an important example. For any field $K$ of characteristic zero, the abelianization of $GL_n(K)$ is $K^{\times}$ since its commutator subgroup is $SL_n(K)$. Furthermore the center is given by $\lbrace\lambda I\mid \lambda \in K^{\times}\rbrace \cong K^{\times}$.
More generally, this holds for Lie groups of the form $S_1\times \cdots \times S_r\times T$ with semisimple $S_i$ and abelian $T$.
Dietrich Burde
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I didn't know this, very interesting. Would you happen to have a reference, I'd love to look at the proof :) – AB_IM Jun 07 '19 at 13:40
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We have $GL_n(\Bbb C)=S\times \Bbb C^{\times}$ and $[S,S]=S$. The center is $\Bbb C^{\times}$. This is all for the proof. For the center see this question. – Dietrich Burde Jun 07 '19 at 13:54