Quasi-K\"ahler Chern-flat manifolds and complex 2-step nilpotent Lie algebras

TL;DR

This paper explores the relationship between quasi-K"ahler Chern-flat manifolds and complex 2-step nilpotent Lie algebras, establishing a correspondence with algebraic structures satisfying specific identities.

Contribution

It establishes a correspondence between geometric structures on manifolds and algebraic structures on Lie algebras, providing new insights and examples in the study of complex nilpotent Lie algebras.

Findings

Quasi-K"ahler Chern-flat structures correspond to complex parallelisable structures satisfying Gray's second identity.

A natural algebraic correspondence between anti-bi-invariant and bi-invariant complex structures is demonstrated.

Several exotic examples of such algebraic structures are described in detail.

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 to complex parallelisable Hermitian structures satisfying the second Gray identity. From an algebraic point of view this correspondence reads as a natural correspondence between anti-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.