Annals of Global Analysis and Geometry

, Volume 53, Issue 4, pp 543–559 | Cite as

Hyper-para-Kähler Lie algebras with abelian complex structures and their classification up to dimension 8

  • Ignacio Bajo
  • Esperanza Sanmartín


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.


Hyper-para-Kähler manifold Abelian (para)complex structure Symplectic left-symmetric algebra 

Mathematics Subject Classification

53C55 53D05 17B30 


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© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Depto. Matemática Aplicada IIE.I. TelecomunicaciónVigoSpain
  2. 2.Depto. MatemáticasFacultad de CC.EEVigoSpain

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