Quasi-Kähler Chern-flat manifolds and complex 2-step nilpotent Lie algebras

Abstract

The study of quasi-K"ahler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras.In the present paper we show that quasi-K"ahler Chern-flat almost Hermitian structures on compact manifolds are in correspondence tocomplex parallelisable Hermitian structures satisfying the second Gray identity.From an algebraic point of view this correspondence reads as a natural correspondence betweenanti-bi-invariant almost complex structures on Lie algebras to bi-invariant complex structures.Some natural algebraic problems are approached and some exotic examples are carefully described.