Assume $G$ is a Lie group. Let $\pi\colon P\to M$ be a principal fiber bundle with structural group $G$ which acts on the right. (Fibers are all isomorphic to $G$.)
Let $C(P,G)$ be the set of all maps $\tau\colon P\to G$ such that $$\tau(pg)=g^{-1}\tau(p).$$ One can replace $G$ in $C(P,G)$ by a vector space $V$ such that $G\to \mathrm{GL}(V)$ is given representation.
Is this $C(P,G)$ a Lie group?
The reason I am asking this is due to the following facts. Denote $GA(P)$ the group of gauge transformations of $P$, i.e. $f\in GA(P)$ iff all the following hold:
- $f\colon P\to P$ is an automorphism,
- $f$ descends to $M$ as $\mathrm{Id}_M$,
- $f(pg)=f(p)g$.
Then $GA(P)^{op}\cong C(P,G)$ are isomorphic as groups (where $op$ means opposite group).
Furthermore, let $\mathfrak{g}$ to be the Lie algebra of $G$. There is an "exponential map" $\overline{\exp}\colon C(P,\mathfrak g)\to C(P,G)$. My guess is that $C(P,\mathfrak g)$ is the tangent space of $C(P,G)$.