Eight-Dimensional Hermitian Lie Groups Conformally Foliated by Minimal $\mathbf{\text{SU}}(2)\times \mathbf{\text{SU}}(2)$ Leaves

We investigate the $8$-dimensional Riemannian Lie groups $G^8$, carrying a left-invariant, conformal and minimal foliation $\mathcal{F}$, with leaves diffeomorphic to the subgroup $\mathbf{\text{SU}}(2)\times \mathbf{\text{SU}}(2)$ of $G^8$. Such groups have been classified by E. Ghandour, S. Gudmundsson and T. Turner in their recent work. They show that these $8$-dimensional Lie groups form a real $13$-dimensional family. For each left-invariant Hermitian structure $J_{\mathcal{V}}$ on $\mathbf{\text{SU}}(2)\times \mathbf{\text{SU}}(2)$, we extend this to an almost Hermitian structures $J$ on $G$ adapted to the foliation $\mathcal{F}$ i.e. respecting the leaf structure on $G$ induced by $\mathcal{F}$. We then classify those $8$-dimensional Lie groups $G$ for which the almost Hermitian structures $J$ are integrable (${\mathcal{W}}_3\oplus {\mathcal{W}}_4$), semi-K"ahler (${\mathcal{W}}_4$), locally conformal K"ahler (${\mathcal{W}}_3$) or even K"ahler ($\mathcal{K}$). It turns out that for each $J_{\mathcal{V}}$ we obtain a $9$-dimensional family of Lie groups $G$ for which $J$ is integrable. In the case of $J$ being semi-K"ahler, we yield a $3$-dimensional family of such groups. We then show that in the cases of $J$ being K"ahler or locally conformal K"ahler there are no solutions i.e. such $8$-dimensional Lie groups do not exist.