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

Abstract

Hyper-para-Kahler 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-Kahler 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-Kahler structures.