Motivated by this question we ask:
Is there any relation between the Ricci curvature of a Lie group and the killing form of its Lie algebra?Under what conditions, they are proportional to each other?
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asked Jun 14, 2016 at 10:06
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4$\begingroup$ See Section 7 of Milnor's "Curvatures of Left Invariant Metrics on Lie Groups": sciencedirect.com/science/article/pii/S0001870876800023 $\endgroup$– Travis WillseCommented Jun 14, 2016 at 13:07
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1 Answer
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When the Lie group is compact and semisimple, then the Killing form $B$ is negatively definite and $-B$ defines a bi-invariant metric. Its Ricci curvature is $$Ric(X)=-\frac{1}{4}\sum_{i=2}^nB(\left[X,e_i\right],\left[X,e_i\right])$$ for any $B$-orthonormal base $\left\{e_1=X,e_2,\ldots,e_n\right\}$ with $e_1=X$. In particular, $Ric(X)$ is nonnegative.
