Transversely K\"ahler almost contact metric Lie algebras

We study transversely K\"ahler almost contact metric Lie algebras $(\mathfrak{g},\varphi,\xi,\eta,g)$ such that the structure $1$-form $\eta$ is a contact form. They include both quasi Sasakian and anti-quasi-Sasakian Lie algebras of maximal rank. In the case where the center of the Lie algebra is nontrivial, they are $1$-dimensional central extensions of K\"ahler Lie algebras via a symplectic form. We investigate the $5$-dimensional case, obtaining a classification of $\eta$-Einstein transversely K\"ahler almost contact metric Lie algebras of maximal rank. If the center is trivial, the structure is always $\alpha$-Sasakian. If the center is nontrivial and the K\"ahler quotient $\mathfrak{g}/\mathfrak{z(g)}$ is not abelian, the structure is quasi Sasakian; it is $\alpha$-Sasakian on central extensions of K\"ahler-Einstein $4$-dimensional Lie algebras, and not conversely. Up to isomorphisms, the Heisenberg Lie algebra $\mathfrak{h}_5$ is the only $5$-dimensional Lie algebra admitting $\eta$-Einstein transversely K\"ahler structures which are not quasi Sasakian, including anti-quasi-Sasakian structures. In fact, we show that any $5$-dimensional anti-quasi-Sasakian Lie algebra is isomorphic to $\mathfrak{h}_5$.