Abstract
Hyper-para-Kähler structures on Lie algebras where the complex structure is abelian are studied. We show that there is a one-to-one correspondence between such hyper-para-Kähler Lie algebras and complex commutative (hence, associative) symplectic left-symmetric algebras admitting a semilinear map \(K_s\) verifying certain algebraic properties. Such equivalence allows us to give a complete classification, up to holomorphic isomorphism, of pairs \(({\mathfrak g},J)\) of 8-dimensional Lie algebras endowed with abelian complex structures which admit hyper-para-Kähler structures.
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Bajo, I., Sanmartín, E. Hyper-para-Kähler Lie algebras with abelian complex structures and their classification up to dimension 8. Ann Glob Anal Geom 53, 543–559 (2018). https://doi.org/10.1007/s10455-017-9587-8
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DOI: https://doi.org/10.1007/s10455-017-9587-8