§ Lie bracket commutator as infinitesimal conjugation
- Consider the map . Say we want to study the map near the identity in the first argument.
- So we replace by for identity and arbitrary group element.
- This now makes conjugation .
- Since by taylor expansion, the above becomes: .
- Algebra time:
- Thus, the linear part/gradient of the conjugation is given by .
- So, the lie bracket corresonds to infinitesimal conjugation.