This paper studies how many orthogonal bi-invariant complex structures exist on a metric Lie algebra over the real numbers. Recently, it was shown that irreducible Lie algebras which are additionally $2$-step nilpotent admit at most one orthogonal bi-invariant complex structure up to sign. The main result generalizes this statement to metric Lie algebras with any number of irreducible factors and which are not necessarily $2$-step nilpotent. It states that there are either $0$ or $2^k$ such complex structures, with $k$ the number of irreducible factors of the metric Lie algebra. The motivation for this problem comes from differential geometry, for instance to construct non-parallel Killing-Yano $2$-forms on nilmanifolds or to describe the compact Chern-flat quasi-Kahler manifolds. The main tool we develop is the unique orthogonal decomposition into irreducible factors for metric Lie algebras with no non-trivial abelian factor. This is a generalization of a recent result which only deals with nilpotent Lie algebras over the real numbers. Not only do we apply this fact to describe the orthogonal bi-invariant complex structures on a given metric Lie algebra, but it also gives us a method to study different inner products on a given Lie algebra, computing the number of irreducible factors and orthogonal bi-invariant complex structures for varying inner products.

中文翻译：

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度量李代数上的正交双不变复结构

本文研究了实数上的度量李代数上存在多少正交双不变复结构。最近，已经表明另外$2$-step 幂零的不可约李代数最多承认一个正交双不变复数结构直到符号。主要结果将这个陈述推广到用任意数量的不可约因子度量李代数，这些因子不一定是 $2$-step 幂零。它指出存在 $0$ 或 $2^k$ 这样的复杂结构，其中 $k$ 是度量李代数的不可约因数的数量。这个问题的动机来自微分几何，例如在 nilmanifolds 上构造非平行的 Killing-Yano $2$-形式或描述紧凑的 Chern-flat quasi-Kahler 流形。我们开发的主要工具是对没有非平凡阿贝尔因子的度量李代数的不可约因子进行独特的正交分解。这是最近结果的推广，该结果仅涉及实数上的幂零李代数。我们不仅应用这个事实来描述给定李代数上的正交双不变复结构，而且还为我们提供了一种研究给定李代数上不同内积的方法，计算不可约因子的数量和正交双- 不同内积的不变复杂结构。