Killing symmetries of deformed relativity in five dimensions

This is the first of two papers devoted to investigating the main mathematical aspects of the Kaluza-Klein-like scheme known as Deformed Relativity in five dimensions (DR5). It is based on a five-dimensional Riemannian space in which the four-dimensional space-time metric is deformed (i.e. it depends on the energy) and energy plays the role of the fifth dimension. After a brief survey of the physical and mathematical foundations of DR5, we discuss in detail the Killing symmetries of the theory. In particular, we consider the case of physical relevance in which the metric coefficients are power functions of the energy (Power Ansatz). In order to solve the related Killing equations, we introduce a simplifying hypothesis of functional independence (Υ hypothesis). The explicit expressions of the Killing vectors for the energydependent metrics corresponding to the four fundamental interactions (electromagnetic, weak, strong and gravitational) are derived. A preliminary discussion of the infinitesimal-algebraic structure of the Killing symmetries of DR5 is also given.