This question is based on the exercise 10.1 (b) in "Geometry, Topology, and Physics" by Nakahara. Let $G\rightarrow P \rightarrow M$ be a principal bundle. Given an element of the Lie algebra $A\in\mathfrak{g}$, we can define a vector $A^{*} \in T_u P$ by its action on a smooth function $f:M\rightarrow \mathbb{R}$ as $A^{*} f(u)=\frac{d}{dt}f(u\cdot \exp(tA) )|_{t=0}$ using the right action of $G$ on $P$. Since $u\cdot\exp(tA)$ defines a flow on $P$, this definition extends to give us a vector field.
I would now like to show that this construction is compatible with the lie bracket, i.e. $[A^{*},B^{*}]=[A,B]^{*}$. My idea was to compute the action of $[A^{*},B^{*}]$ on a function $f$, as above. However, I get $A^{*}B^{*}f(u)=\frac{\partial}{\partial t}|_{t=0}\frac{\partial}{\partial s}|_{s=0}f(u\cdot\exp(sB)\exp(tA))=\frac{\partial}{\partial t}|_{t=0}\frac{\partial}{\partial s}|_{s=0}f(u\cdot\exp(sB+tA+\frac{st}{2}[B,A]+(s^3,t^3...)))$. Repeating this calculation for $B^{*}A^{*}f(u)$, expanding the resulting expressions in $s$ and $t$, and combining to get the commutator then seems to give $[A^{*},B^{*}]=[B,A]^{*}$ which is incorrect.
My questions are:
- Where is my mistake leading to the incorrect sign for the commutator?
- Does it make sense to expand in $s$ and $t$? I get confused when trying to do so carefully because I inevitably end up mixing elements in $G$ and elements in $\mathfrak{g}$.
Many thanks for your help.