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Locally conformally Kähler structures on unimodular Lie groups

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Abstract

We study left-invariant locally conformally Kähler structures on Lie groups, or equivalently, on Lie algebras. We give some properties of these structures in general, and then we consider the special cases when its complex structure is bi-invariant or abelian. In the former case, we show that no such Lie algebra is unimodular, while in the latter, we prove that if the Lie algebra is unimodular, then it is isomorphic to the product of \(\mathbb {R}\) and a Heisenberg Lie algebra.

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Acknowledgments

The authors would like to thank L. Ornea and I. Dotti for useful comments and the referee for his/her thorough review of the manuscript and the constructive comments. The authors were partially supported by CONICET, ANPCyT and SECyT-UNC (Argentina).

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Correspondence to M. Origlia.

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Andrada, A., Origlia, M. Locally conformally Kähler structures on unimodular Lie groups. Geom Dedicata 179, 197–216 (2015). https://doi.org/10.1007/s10711-015-0076-6

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