Left invariant special K\"ahler structures

We construct left invariant special K\"ahler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special K\"ahler Lie algebras according to two linear representations by infinitesimal K\"ahler transformations. We also exhibit a double extension process of a special K\"ahler Lie algebra which allows us to get all simply connected special K\"ahler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or K\"ahler) affine transformations of its Lie algebra containing a nontrivial $1$-parameter subgroup formed by central translations. We show a characterization of left invariant flat special K\"ahler structures using \'etale K\"ahler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.