How Does the Adjoint Map Function in Lie Theory?

[Mandelbroth]

I'm trying to delve a little deeper into using Lie groups and Lie algebras. Right now, I'm wondering if there's an optimal way to first consider the adjoint map (derivation).

Right now, I'm trying to get comfortable with Lie algebras, so I'm thinking it's best to play the role of the mathematical idiot and not acknowledge that there is a connection between the adjoint map (derivation) and the Adjoint map (automorphism).

Does anyone concur with this ideology?

 

[jgens]

Either approach works. I first learned Lie algebra representations because they provide useful information about Lie group representations and most of the material was motivated by this connection. This is the approach taken in books like Fulton and Harris. Alternatively you could just focus on Lie algebra representations on their own and this is the approach taken in books like Humphreys.

 

[Mandelbroth]

Either approach works. I first learned Lie algebra representations because they provide useful information about Lie group representations and most of the material was motivated by this connection. This is the approach taken in books like Fulton and Harris. Alternatively you could just focus on Lie algebra representations on their own and this is the approach taken in books like Humphreys.

That's actually how this question came up. I'm using my copy of Humphreys for this, and I thought that it was odd to not mention the connection.

Thank you again, jgens.

 

[jgens]

That's actually how this question came up. I'm using my copy of Humphreys for this, and I thought that it was odd to not mention the connection.

Nah. Once you finish Humphreys definitely read a book on Lie group representations (Fulton and Harris or Knapp would be my recommendations), but for a first fun through the material his approach is fine.