We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being “nice”. Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Lauret and Will (Proc Am Math Soc 141(10):3651–3663, 2013) on nilpotent Lie algebras. The result follows from a more general characterization for diagonalizing the Ricci tensor for homogeneous spaces. Finally, we also study the Ricci flow behavior of diagonal metrics on cohomogeneity one manifolds.

中文翻译：

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对角化Ricci张量

我们证明了紧凑型半简单李代数的基础，对于它，任何对角左不变度量都具有对角Ricci张量，其特征在于李代数条件为“ nice”。即，任何两个基本元素的括号是另一个基本元素的倍数。这扩展了Lauret和Will（Proc Am Math Soc 141（10）：3651-3663，2013）在幂等李代数上的工作。该结果来自对均匀空间的Ricci张量进行对角化的更一般的表征。最后，我们还研究了同构一流形上对角度量的Ricci流行为。