We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $$\le 6$$ ≤ 6 , every nice nilpotent Lie group of dimension $$\le 7$$ ≤ 7 and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups $$\mathrm {SL}(n)$$ SL ( n ) , $$\mathrm {SO}(p,q)$$ SO ( p , q ) , $$\mathrm {Sp}(n,\mathbb {R})$$ Sp ( n , R ) . Most of these metrics are shown not to be flat.

中文翻译：

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图对合和齐次 Ricci-flat 度量

我们引入了一种组合方法来在好的幂零李群上构造不定 Ricci-flat 度量。我们证明了每一个维度 $$\le 6$$ ≤ 6 的幂零李群，每一个维度 $$\le 7$$ ≤ 7 的好的幂零李群和每个附加到图上的两步幂零李群都承认这样一个公制。我们在简单李群 $$\mathrm {SL}(n)$$ SL ( n ) , $$\mathrm {SO}(p,q)$$ SO ( p , q ) , $$\mathrm {Sp}(n,\mathbb {R})$$ Sp ( n , R ) 。这些指标中的大多数都不是平坦的。