nLab automorphism group of a Lie algebra

Definition

For $\mathfrak{g}$ a Lie algebra, its automorphism group $Aut(\mathfrak{g}) \subset GL(\mathfrak{g})$ (in the category LieAlg) is the subgroup of the general linear group of invertible linear maps $\alpha \colon \mathfrak{g} \xrightarrow{\sim} \mathfrak{g}$ which preserve the Lie bracket, in that for $\alpha \in Aut(\mathfrak{g})$ and $x,y\in \mathfrak{g}$ we have

$$\alpha\big( [x,y] \big) = \big[ \alpha(x), \alpha(y) \big] \,.$$

The group operation is the composition of linear maps.

This group canonically carries the structure of a Lie group. The corresponding Lie algebra is the derivation Lie algebra of $\mathfrak{g}$.

Properties

There is a normal subgroup $Aut(\mathfrak{g})_0$ of automorphisms, the inner automorphisms, obtained via the adjoint action as

$$exp\big(ad(x)\big) \colon \mathfrak{g} \longrightarrow \mathfrak{g} \,,$$

for $x\in \mathfrak{g}$, giving rise to the exact sequence

$$1 \to Aut(\mathfrak{g})_0 \longrightarrow Aut(\mathfrak{g}) \longrightarrow \pi_0\big( Aut(\mathfrak{g}) \big) \to 1 \mathrlap{\,.}$$

The quotient group $\pi_0\big(Aut(\mathfrak{g})\big)$ of outer automorphisms is discrete.

This sequence is known to be a split extension for $\mathfrak{g}$ a simple Lie algebra of finite dimension over the reals or over the complex numbers (Gündogan 2010).

References

• Hasan Gündogan: The component group of the automorphism group of a simple Lie algebra and the splitting of the corresponding short exact sequence, J. Lie Theory 20 4 (2010) 709-737 [jlt:20035, pdf]

Related concepts

• endomorphism Lie algebra

• automorphism Lie algebra

• automorphism of a vertex operator algebra

• automorphism infinity-Lie algebra