A class of Sasakian 5-manifolds

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 of R by H(3)x R. In particular, the compact quotient is an S^1-bundle over a 4-dimensional Kahler solvmanifold.