Short distance expansion from the dual representation of infinite dimensional lie algebras

We develop a method for computing the short distance expansion of fields or operators that live in the coadjoint representation of an infinite dimensional Lie algebra by using only properties of the adjoint representation and its dual. We explicitly implement this method by computing the short distance expansion for the duals of the Virasoro algebra, affine Lie algebras and the geometrically realized N-extended supersymmetric GR Virasoro algebra. This method can also be used to compute short distance expansions between fields that transform in the adjoint and those that transform in the coadjoint representations.