Special Hermitian metrics and Lie groups

A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form ω   is ∂∂¯-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 gg and using a Hermitian flat connection on gg 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.