- Thread starter
- Moderator
- #1

- Jan 26, 2012

- 995

-----

**Problem**: Any linear transformation $\delta$ of a Lie algebra $\mathfrak{g}$ with the property $\delta[Y,Z] = [\delta Y, Z] + [Y,\delta Z]$ is called a

*derivation*of $\mathfrak{g}$. We denote the collection of all derivations of $\mathfrak{g}$ by the set $\text{Der}(\mathfrak{g})$. Show that $\text{Der}(\mathfrak{g})$ is the Lie algebra of the linear group $\text{Aut}(\mathfrak{g})$.

-----

\[\delta[Y,Z] = [\delta Y, Z] + [Y,\delta Z]\iff \exp(\tau\delta)[Y,Z] = [\exp(\tau\delta) Y, Z] + [Y,\exp(\tau\delta) Z].\]

(the second equation tells us that in this case, $\exp(\tau\delta)$ is an automorphism for all $\tau\in\mathbb{R}$).

Remember to read the POTW submission guidelines to find out how to submit your answers!