The conformal factor in the SAS Einstein-Maxwell field equations and a central extension of a formal loop group

We consider a relation between the conformal factor in the stationary axisymmetric (SAS) Einstein-Maxwell field equations and a central extension of a formal loop group which is described by a group 2-cocycle on the formal loop group. The corresponding 2-cocycle on the Lie algebra of the formal loop group is the one which describes an affine Lie algebra. As a result, we see that the space of formal solutions with conformal factors is a homogeneous space of a central extension of the Hauser group.