Universal covering of maximal torus in a compact Lie group

Question

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

You mean $\textbf{maximal torus}$ inside a Lie group? – Bombyx mori Feb 04 '14 at 15:08 — Bombyx mori, Feb 04 '14 at 15:08
I see. I shall venture to correct the typos. – Bombyx mori Feb 04 '14 at 15:19 — Bombyx mori, Feb 04 '14 at 15:19

score 4 · Accepted Answer · answered Feb 04 '14 at 15:09

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.

Universal covering of maximal torus in a compact Lie group

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