Symplectic Lie groups and doubled geometry

Abstract

A left invariant flat para-Kähler structure is constructed on the tangent Lie group of a symplectic Lie group (G, ω). Remarkably, it is shown that the left invariant para-Kähler form on TG coincides with a certain pullback of the standard symplectic form on T∗G. The immediate upshot of this is that T∗G can be equipped with a Lie group structure for which the standard symplectic form is left invariant. Lastly, the double field theory geometry of the double manifold TG is studied using the geometric framework of Vaisman [22, 23].