  We obtain some general results on Sasakian Lie algebras and prove as a
consequence that a (2n + 1)-dimensional nilpotent Lie group admitting
left-invariant Sasakian structures is isomorphic to the real Heisenberg group
$H_{2n + 1}$. Furthermore, we classify Sasakian Lie algebras of dimension 5 and
determine which of them carry a Sasakian $\alpha$-Einstein structure. We show
that a 5-dimensional solvable Lie group with a left-invariant Sasakian
structure and which admits a compact quotient by a discrete subgroup is
isomorphic to either $H_5$ or a semidirect product $\R \ltimes (H_3 \times
\R)$. In particular, the compact quotient is an $S^1$-bundle over a
4-dimensional K\"ahler solvmanifold.
