On the tangent Lie group of a symplectic Lie group

Motivated by the recent work of Asgari and Salimi Moghaddam (Rend Circ Mat Palermo II Ser 67:185-195, 2018) on the Riemannian geometry of tangent Lie groups, we prove that the tangent Lie group TG of a symplectic Lie group (G,omega) admits the structure of a symplectic Lie group. On TG, we construct a left invariant symplectic form (omega) over tilde which is induced from omega using complete and vertical lifts of left invariant vector fields on G. The aforementioned construction can be viewed as the symplectic analogue of the left invariant Riemannian metrics on the tangent Lie groups that were constructed in Asgari and Salimi Moghaddam (Rend Circ Mat Palermo II Ser 67:185-195, 2018). One immediate upshot of our construction is that by taking iterated tangent bundles of a non-abelian symplectic Lie group, one obtains a convenient means of generating non-abelian symplectic Lie groups of arbitrarily high dimension.