Special Hermitian metrics and Lie groups

A Hermitian metric on a complex manifold is called strong K\"ahler with torsion (SKT) if its fundamental 2-form $\omega$ is $\partial \bar \partial$-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a $2n$-dimensional SKT Lie algebra $\mathfrak g$ {and using} a Hermitian flat connection on $\mathfrak g$ we construct a $4n$-dimensional SKT Lie algebra. We apply this method to some 4-dimensional SKT Lie algebras. Moreover, we classify symplectic forms taming complex structures on 4-dimensional Lie algebras.