In this paper we study invariant projectively flat affine connections and invariant dualistic structures of constant curvature. We first relate the existence of invariant projectively flat affine connections to that of certain affine representation of Lie algebras (Theorem 1). Using such affine representations we give a correspondence between semisimple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras (Theorem 2). As an application we prove that invariant dualistic structures of constant curvature come from certain invariant Hessian structures (Theorem 3).