We consider families of linear, parabolic PDEs in n dimensions which possess Liesymmetrygroups of dimension at least four. We identify the Liesymmetrygroups of these equations with the (2n+1)-dimensional Heisenberg group and SL(2,R). We then show that for PDEs of this type, the Liesymmetries may be regarded as global projective representations of the symmetrygroup. We construct explicit intertwining operators between the symmetries and certain classical projective representations of the symmetrygroups. Banach algebras of symmetries are introduced.