Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple $(G,omega, abla)$ such that $G$ is a finite-dimensional real Lie group and $omega$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $ abla$. In this paper starting from a special symplectic Lie group we show how to ``deform the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure $ abla$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.