Balanced Hermitian structures on twisted cartesian products

Abstract

We study Hermitian structures on twisted cartesian products $({\mathfrak{g}}_{(\ddot{I}{}_1,\ddot{I}{}_2)},\mathrm{J},\mathcal{K})$ of two Hermitian Lie algebras according to two representations $\ddot{I}{}_1$ and $\ddot{I}{}_2$. We give the conditions on $({\mathfrak{g}}_{(\ddot{I}{}_1,\ddot{I}{}_2)},\mathrm{J},\mathcal{K})$ to be balanced and locally conformally balanced. As an application we classify six-dimensional balanced Hermitian twisted cartesian products Lie algebras.