Automorphism of a Lie algebra

From HandWiki
Short description: Type of automorphism

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

Examples

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


Theorems

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

References

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