What is the universal covering of maximal torus $T$ of dimension $n$? of a compact Lie group $G$. Is there any reference?

  • $\begingroup$ You mean $\textbf{maximal torus}$ inside a Lie group? $\endgroup$ – Bombyx mori Feb 4 '14 at 15:08
  • $\begingroup$ yes, exactly of compact Lie group $\endgroup$ – user61135 Feb 4 '14 at 15:09
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    $\begingroup$ I see. I shall venture to correct the typos. $\endgroup$ – Bombyx mori Feb 4 '14 at 15:19

The exponential map $\exp:\mathfrak{t}\rightarrow T$ will be the universal cover. (The exponential map for $G$ restricts to the one I have written.) To verify this explicitly, just consider the case $T=U(1)^n$ and $\mathfrak{t}=(i\mathbb{R})^n$. In this case, the exponential map is the usual exponential map in each factor.


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