A Universe of Sorts

§ Lie bracket commutator as infinitesimal conjugation

Consider the map $c(g, h) = ghg^{-1}$ . Say we want to study the map near the identity in the first argument.
So we replace $g$ by $e + \epsilon k$ for identity $e$ and $k$ arbitrary group element.
This now makes conjugation $(e + \epsilon k) h (e + \epsilon k)^{-1}$ .
Since $(1+x)^{-1} \simeq 1 - x$ by taylor expansion, the above becomes: $(e + \epsilon k) h (e - \epsilon k)$ .
Algebra time:

\begin{aligned} &(e + \epsilon k) h (e - \epsilon k) \\ &= ehe + e h (- \epsilon k) + \epsilon k) h e - \epsilon k h (-\epsilon k) \\ &= h - \epsilon hk + \epsilon kh - \epsilon^2 khk \\ &\simeq h + \epsilon [k, h] + O(\epsilon^2) \end{aligned}

Thus, the linear part/gradient of the conjugation is given by $\epsilon [k, h] = kh - hk$ .
So, the lie bracket corresonds to infinitesimal conjugation.