Flat Hermitian Lie algebras are K\"ahler

Abstract

In 1976, Milnor classified all Lie groups admitting a flat left-invariant metric. They form a special type of unimodular 2-step solvable groups. Considering Lie groups with Hermitian structure, namely, a left-invariant complex structure and a compatible left-invariant metric, in 2006, Barberis-Dotti-Fino obtained among other things full classification of all Lie groups with Hermitian structure that are K\"ahler and flat. In this note, we examine Lie groups with a Hermitian structure that are flat, and show that they actually must be K\"ahler, or equivalently speaking, a flat Hermitian Lie algebra is always K\"ahler. In the proofs we utilized analysis on the Hermitian geometry of 2-step solvable Lie groups developed by Freibert-Swann and by Chen and the second named author.