论文标题
公制谎言代数上的正交双反变体结构
Orthogonal bi-invariant complex structures on metric Lie algebras
论文作者
论文摘要
本文研究了在实数方面，在度量代数上存在多少个正交的双重不变结构。最近，结果表明，不可约合的lie代数额外$ 2 $ step nilpotent允许最多可以签名的正交双性异性复合体结构。主要结果将此陈述概括为公制代数，并具有许多不可约的因素，并且不一定是$ 2 $ - 步骤的nilpotent。它指出，这种复杂的结构有$ 0 $或$ 2^k $，其中$ k $是公制Lie代数的不可约因素的数量。这个问题的动机来自差异几何形状，例如，在尼尔曼福德（Nilmanifolds）上构建非并行杀死yano $ 2 $ - 形式或描述紧凑的Chern-Flat Quasi-Kähler歧管。 我们开发的主要工具是独特的正交分解为不可约束的代数，而没有非平凡的阿贝尔因子。这是对最近结果的概括，该结果仅涉及nilpotent lie代数在实际数字上。我们不仅应用了这一事实来描述给定的谎言代数上的正交双重不变的复合结构，而且还为我们提供了一种在给定的谎言代数上研究不同内部产品的方法，计算了不可约束的因素和正交双性异性复杂结构的数量。
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-Kähler 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.
