ABSTRACT
In the geometric space of dimension, a referential of orthogonal axes, a point is located in relation to the origin by its coordinates. One can write the elements of an orthogonal matrix in terms of the elements of a homomorphic matrix. An operator that commutes with all the generators is called the Casimir operator. The Casimir operator is important in quantum mechanics for the construction of a complete set of observables that commute, as well as being important in the research on grand unified theories. The number of Casimir operators, for a given group, defines the rank of the group. A quadrivector is said to be transformed by a nonhomogeneous element of a Poincare or Lorentz group if its components are such that the first term of the right-hand member is associated with a Lorentz transformation, whereas the second term is a translation of four dimensions.
