The study of quasi-Kähler 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-Kähler 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 and bi-invariant complex structures. Some natural algebraic problems are approached and some exotic examples are carefully described.
@article{ASNSP_2012_5_11_1_41_0,
author = {Di Scala, Antonio J. and Lauret, Jorge and Vezzoni, Luigi},
title = {Quasi-K\"ahler {Chern-flat} manifolds and complex $2$-step nilpotent {Lie} algebras},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {41--60},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {1},
year = {2012},
mrnumber = {2953044},
zbl = {1252.53038},
language = {en},
url = {http://archive.numdam.org/item/ASNSP_2012_5_11_1_41_0/}
}
TY - JOUR AU - Di Scala, Antonio J. AU - Lauret, Jorge AU - Vezzoni, Luigi TI - Quasi-Kähler Chern-flat manifolds and complex $2$-step nilpotent Lie algebras JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2012 SP - 41 EP - 60 VL - 11 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2012_5_11_1_41_0/ LA - en ID - ASNSP_2012_5_11_1_41_0 ER -
%0 Journal Article %A Di Scala, Antonio J. %A Lauret, Jorge %A Vezzoni, Luigi %T Quasi-Kähler Chern-flat manifolds and complex $2$-step nilpotent Lie algebras %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2012 %P 41-60 %V 11 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2012_5_11_1_41_0/ %G en %F ASNSP_2012_5_11_1_41_0
Di Scala, Antonio J.; Lauret, Jorge; Vezzoni, Luigi. Quasi-Kähler Chern-flat manifolds and complex $2$-step nilpotent Lie algebras. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 41-60. http://archive.numdam.org/item/ASNSP_2012_5_11_1_41_0/
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