The components of adjoint representations are matrices or operators that represent the action of a Lie algebra on itself. They describe how the basis elements of the algebra are transformed under the action of the algebra.
Adjoint representations are important in physics because they describe the symmetries of a physical system. These symmetries can be used to identify conserved quantities, such as energy and momentum, and to study the dynamics of the system.
Adjoint representations are closely related to Lie algebras, as they are representations of the Lie algebra on itself. This means that the matrices or operators that make up the adjoint representation are elements of the Lie algebra.
Yes, adjoint representations can be used to study non-Lie groups as long as the group has a Lie algebra associated with it. In this case, the adjoint representation acts on the Lie algebra, rather than the group itself.
Adjoint representations are related to the concept of duality in that they can be thought of as the "dual" of the fundamental representation. In other words, while the fundamental representation describes the action of the group on its vectors, the adjoint representation describes the action of the group on its own structure.