An automorphism is a homomorphism of a group $G$ onto itself. For Lie groups this induces a Lie algebra $g$ automorphism, i.e. a map of the Lie alegbra onto itself that preserves the Lie bracket.

An inner (=trivial) automorphism a map that satisfies additionally

$$ \tilde T_a = R T_a R^{-1}, $$

where $T_a \in g$ denotes the generators of the group (= the Lie algebra elements) and $R= e^{i \alpha_a T_a} \in G$ denotes a group element.

In other words, an inner automorphism simply means that we act with group elements on the Lie algebra elements:

$$ G \circ T_a = R T_a R^{-1} $$

We can write each Lie algebra representation in terms of weights and for the adjoint representation the weights are called roots.

How can I compute how the generators change under inner automorphisms in terms of roots and weights? In other words: How can $ G \circ T_a = R T_a R^{-1}$ be transferred into the language of roots and weights?


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