Solving for a Lie Algebra in General

[pdxautodidact]

So,

I just went through the derivation of the Lie algebra for SO(n). in order to do so, we considered $b^{-1}ab$, and related it to $U\left(b^{-1}ab\right)$, and since we have a group homomorphism, $U^{-1}\left(b\right)U\left(a\right)U\left(b\right)$, all of which correspond to the whole similarity matrix thing. By careful choice of element representation, one is able to massage a commutator structure, then it all reduces down. What is the deal with the choice of similarity? Is this why mathematicians always say in passing how important similarity transformations are for physicists?

cheers

 

[fresh_42]

We are in need of an operation by the Lie group on its tangent space, and conjugation is the natural choice, as it comes from the conjugation in the group - an operation every group naturally has. Compare:
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-basics/