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Hypercomplex Almost Abelian Solvmanifolds

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Abstract

We give a characterization of almost abelian Lie groups carrying left invariant hypercomplex structures and we show that the corresponding Obata connection is always flat. We determine when such Lie groups admit HKT metrics and study the corresponding Bismut connection. We obtain the classification of hypercomplex almost abelian Lie groups in dimension 8 and determine which ones admit lattices. We show that the corresponding 8-dimensional solvmanifolds are nilmanifolds or admit a flat hyper-Kähler metric. Furthermore, we prove that any 8-dimensional compact flat hyper-Kähler manifold is a solvmanifold equipped with an invariant hyper-Kähler structure. We also construct almost abelian hypercomplex nilmanifolds and solvmanifolds in higher dimensions.

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Notes

  1. A solvable Lie group G is completely solvable if the adjoint operators \({\text {ad}}_x:{\mathfrak {g}}\rightarrow {\mathfrak {g}}\), with \(x\in {\mathfrak {g}}={\text {Lie}}(G)\), have only real eigenvalues. In particular, nilpotent Lie groups are completely solvable.

  2. Two hypercomplex structures \(\{J_{\alpha }\}\) and \(\{J_\alpha '\}\) on a Lie algebra \({\mathfrak {g}}\) are equivalent if there exists a Lie algebra automorphism \(\psi \) of \({\mathfrak {g}}\) such that \(\psi J_\alpha =J_\alpha '\psi \) for all \(\alpha \).

  3. For square matrices X, Y, we denote \(X\oplus Y=\begin{bmatrix} X&{}\quad 0 \\ 0&{}\quad Y \end{bmatrix}\). We use a similar notation for 3 or more matrices.

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Acknowledgements

The authors are grateful to Leandro Cagliero, Graziano Gentili, Giulia Sarfatti and Alejandro Tolcachier for useful comments.

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  1. FAMAF, Universidad Nacional de Córdoba and CIEM-CONICET, Av. Medina Allende s/n, Ciudad Universitaria, X5000HUA, Córdoba, Argentina

    Adrián Andrada & María Laura Barberis

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  1. Adrián Andrada
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  2. María Laura Barberis
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Correspondence to Adrián Andrada.

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Andrada, A., Barberis, M.L. Hypercomplex Almost Abelian Solvmanifolds. J Geom Anal 33, 213 (2023). https://doi.org/10.1007/s12220-023-01277-y

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