# Automorphism of a Lie algebra

Short description: Type of automorphism

In abstract algebra, an automorphism of a Lie algebra $\displaystyle{ \mathfrak g }$ is an isomorphism between $\displaystyle{ \mathfrak g }$ and itself; i.e., a linear automorphism that preserves the bracket. The totality of them forms the automorphism group $\displaystyle{ \operatorname{Aut}(\mathfrak g) }$ of $\displaystyle{ \mathfrak g }$. The subgroup of $\displaystyle{ \operatorname{Aut}(\mathfrak g) }$ generated by matrix exponents $\displaystyle{ e^{\operatorname{ad}(x)}, x \in \mathfrak g }$ is called the inner automorphism group of $\displaystyle{ \mathfrak g }$.

## Examples

• For each $\displaystyle{ g }$ in a Lie group $\displaystyle{ G }$, let $\displaystyle{ \operatorname{Ad}_g }$ denote the differential at the identity of the conjugation by $\displaystyle{ g }$. Then $\displaystyle{ \operatorname{Ad}_g }$ is an automorphism of $\displaystyle{ \mathfrak{g} = \operatorname{Lie}(G) }$, the adjoint action by $\displaystyle{ g }$.

## Theorems

The Borel–Morozov theorem states that every solvable subalgebra of a complex semisimple Lie algebra $\displaystyle{ \mathfrak g }$ can be mapped to a subalgebra of a Cartan subalgebra $\displaystyle{ \mathfrak h }$ of $\displaystyle{ \mathfrak g }$ by an inner automorphism of $\displaystyle{ \mathfrak g }$. In particular, it says that $\displaystyle{ \mathfrak h \oplus \oplus_{\alpha \gt 0} \mathfrak{g}_{\alpha} }$, where $\displaystyle{ \mathfrak{g}_{\alpha} }$ are root spaces, is a maximal solvable subalgebra (i.e., a Borel subalgebra).[1]

## References

1. Serre 2000, Ch. VI, Theorem 5.