  A special symplectic Lie group is a triple $(G,\omega,\nabla)$ 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 $\nabla$. 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 $\nabla$ 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.
