Flat symplectic Lie algebras

Let (G,Ω) be a symplectic Lie group, i.e., group endowed with left invariant form. If g is the algebra of G then we call (g,ω=Ω(e))a algebra. The product • on defined by 3ω(x•y,z)=ω([x,y],z)+ω([x,z],y) extends to connection ∇ which torsion free and (∇Ω=0). When has vanishing curvature, (G,Ω)a flat (g,ω)a In this paper, study groups. We start showing that derived ideal degenerate respect ω. show must nilpotent center. This implies always complete. prove double extension process can applied characterize all algebras. More precisely, every obtained sequence algebras starting from {0}. As examples in low dimensions, classify dimension ≤6.