Automorphism of a Lie algebra

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

Examples

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

See also: Cartan involution

Theorems

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

References

• E. Cartan, Le principe de dualité et la théorie des groupes simples et semi-simples. Bull. Sc. math. 49, 1925, pp. 361–374.
• Serre, Jean-Pierre (2000), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4.