Exponentiation of infinite dimensional Z-graded Lie algebras

In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the space V naturally to a representation on a subspace L of the dual space VL. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and affine Kac-Moody Lie algebras.