  The Hessian geometry is the real analogue of the K\"ahler one. Sasakian
geometry is an odd-dimensional counterpart of the K\"ahler geometry. In the
paper, we study the connection between projective Hessian and Sasakian
manifolds analogous to the one between Hessian and K\"ahler manifolds. In
particular, we construct a Sasakian structure on $TM\times \mathbb{R}$ from a
projective Hessian structure on $M$. Especially, we are interested in the case
of invariant structure on Lie groups. We define semi-Sasakian Lie groups as a
generalization of Sasakian Lie groups. Then we construct a semi-Sasakian
structure on a group $G\ltimes \mathbb{R}^{n+1}$ for a projective Hessian Lie
group $G$. Further, we describe examples of homogeneous Hessian Lie groups and
corresponding semi-Sasakian Lie groups. The big class of projective Hessian Lie
groups can be constructed by homogeneous regular domains in $\mathbb{R}^n$. The
groups $\text{SO}(2)$ and $\text{SU}(2)$ belong to another kind of examples.
Using them, we construct semi-Sasakian structures on the group of the Euclidean
motions of the real plane and the group of isometries of the complex plane.
