[[Lie algebras MOC]]
# Lie algebra automorphism

Let $\mathfrak{g}$ be a [[Lie algebra]]. An **[[automorphism]]** $\varphi \in \Aut \mathfrak{g}$ is an [[Lie algebra isomorphism|isomorphism]] $\varphi : \mathfrak{g} \to \mathfrak{g}$. #m/def/lie 
In particular,

1. If $\mathfrak{g} \leq \mathfrak{gl}(V)$, then $\opn{GL}(V) \leq \Aut \mathfrak{g}$.
2. The [[Exponential of a derivation on a Lie algebra]] is an automorphism.
3. Applying the above to the [[Adjoint Lie algebra representation|adjoint representation]] yields [[Inner Lie algebra automorphism|Inner automorphisms]] $\Inn \mathfrak{g} \trianglelefteq \Aut \mathfrak{g}$.

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#state/tidy | #lang/en | #SemBr