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.

 

[blue_raver22]

As a scientist with a background in mathematics, I can say that your approach of first focusing on the adjoint map as a derivation is a valid and important step in understanding Lie groups and Lie algebras. The adjoint map as a derivation is a fundamental concept in Lie theory and plays a crucial role in many applications.

However, it is also important to eventually recognize the connection between the adjoint map as a derivation and the Adjoint map as an automorphism. This connection allows for a deeper understanding of the structure and properties of Lie groups and Lie algebras.

In summary, it is beneficial to first focus on the adjoint map as a derivation in order to build a solid foundation in Lie theory, but eventually acknowledging the connection to the Adjoint map as an automorphism will enhance your understanding and applications of this powerful mathematical tool.