ABSTRACT
Lie groups and Lie algebras, which capture certain symmetries of interest, are rather abstract mathematical objects. Given a Lie group or algebra, a representation of it yields objects that satisfy that symmetry from the outset. In this sense, a representation makes a symmetry more concrete. The adjoint representation has a dimension equal to the dimension of the Lie group and Lie algebra. For our applications in physics, people are interested in tensor fields that display a specific transformation behavior under basic symmetry transformations of interest. Technically, the tensor fields are viewed as functions of discrete tensorial indices and continuous spacetime indices. The isometry transformations, in turn, constitute a Lie group. The reasoning in the conformal case is done in the same way as in the isometric case. Representation theory is a vast field with many applications, and have covered only the basics necessary for our purposes.
