# Monday, August 22

## Humphreys 2.3: Automorphisms

:::{.remark}
Let $L\in \Lie\Alg\slice\FF$, then $\Aut(L)$ is the group of isomorphisms $L \iso L$.
Some important examples: if $L$ is linear and $g\in \GL(V)$, if $gLg\inv = L$ then $x\mapsto gxg\inv$ is an automorphism.
This holds for example if $L = \liegl_n(\FF)$ or $\liesl_n(\FF)$.
Assume $\characteristic \FF = 0$ and let $x\in L$ with $\ad x$ nilpotent, say $(\ad x)^k=0$.
Then the power series expansion $e^{\ad x} = \sum_{n\geq 0} (\ad x)^n$ is a polynomial.
:::

:::{.claim}
$\exp^{\ad x}\in \Aut(L)$ is an automorphism.
More generally, $e^\delta\in \Aut(L)$ for $\delta$ any nilpotent derivation.
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:::{.lemma title="Generalized Leibniz rule"}
\[
\delta^n(xy) = \sum_{i=0}^n {n\choose i} \delta^{n-i}(x) \delta^{i}(y)
.\]
:::

:::{.remark}
One can prove this by induction.
Then check that $\exp(\delta(x))\exp(\delta(y)) = \exp(\delta(xy))$ and writing $\exp(\delta) = 1+\eta$ there is an inverse $1-\eta +\eta^2 +\cdots \pm \eta^{k-1}$.
Automorphisms which are of the form $\exp(\delta)$ for $\delta$ nilpotent derivation are called **inner automorphisms**, and all others are **outer automorphisms**.
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