Automorphisms of Lie Algebras
Sorry for the delay; I’ve had a couple busy days. Here’s Thursday’s promised installment.
An automorphism of a Lie algebra is, as usual, an invertible homomorphism from onto itself, and the collection of all such automorphisms forms a group .
One obviously useful class of examples arises when we’re considering a linear Lie algebra . If is an invertible endomorphism of such that then the map is an automorphism of . Clearly this happens for all in the cases of and the special linear Lie algebra — the latter because the trace is invariant under a change of basis.
Now we’ll specialize to the (usual) case where no multiple of is zero, and we consider an for which is “nilpotent”. That is, there’s some finite such that — applying sufficiently many times eventually kills off every element of . In this case, we say that itself is “ad-nilpotent”.
In this case, we can define . How does this work? we use the power series expansion of the exponential:
We know that this series converges because eventually every term vanishes once .
Now, I say that . In fact, while this case is very useful, all we need from is that it’s a nilpotent derivation of . The product rule for derivations generalizes as:
So we can write
That is, preserves the multiplication of the algebra that is a derivation of. In particular, in terms of the Lie algebra , we find that
Since we conclude that this is an epimorphism of . It’s invertible by the usual formula
which means it’s an automorphism of .
Just like a derivation of the form is called inner, an automorphism of the form is called an inner automorphism, and the subgroup they generate is a normal subgroup of . Specifically, if and then we can calculate
so the conjugate of an inner automorphism is again inner.