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If $G$ is a semi-simple Lie group and $g\in G$, then $G$ has a bi-invariant metric which is a Riemmanian metric. My question is:

with respect to this metric does the left-translation map \begin{equation} h \mapsto g\cdot h \end{equation} define an isometry on $G$?

(I apologize in advance if this question is trivial).

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This is almost tautological. "Bi-invariant" means the metric is invariant under both left and right translation. The resulting distance function satisfies, in particular, $$d(gx,gy)=d(x,y)$$ Hence left translation is an isometry (as is right translation).

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  • $\begingroup$ Good point, sorry about that. And thanks for the answer :) $\endgroup$ – AIM_BLB Jan 28 '16 at 0:50
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    $\begingroup$ @CSA no problem. $\endgroup$ – Matt Samuel Jan 28 '16 at 0:50

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