Abstract
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.
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Acknowledgements
I would like to thank Adrián Andrada for introducing me to Killing-Yano forms during my research stay at Universidad Nacional de Córdoba, which lead to this work. I am grateful to the referee for his/her careful reading of the manuscript.
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The author was supported by a postdoctoral fellowship of the Research Foundation – Flanders (FWO). Note that this manuscript has no associated data.
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Deré, J. Orthogonal bi-invariant complex structures on metric Lie algebras. Ann Glob Anal Geom 59, 157–177 (2021). https://doi.org/10.1007/s10455-020-09746-1
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DOI: https://doi.org/10.1007/s10455-020-09746-1