ABSTRACT
Some automorphisms are trivial in the sense that the mapping they induce on the generators is an equivalence. This is called an inner automorphism. But some of the Lie algebras have non-trivial or outer automorphisms. Since the representations are complex conjugates of one another, this is just the automorphism induced by complex conjugation, up to some trivial equivalence. All of the complex conjugation automorphisms are obtained in this way, associated with reflection symmetries of the Dynkin diagram. The reflection symmetries of their Dynkin diagrams correspond to nontrivial automorphisms that are not complex conjugations. The automorphisms map the representations into one another in all possible ways.
