While reading through Brylinski, as in all of my posts, I am trying to understand the following equation:

$ g_* \tilde{\theta} = \tilde{\theta} - g^{-1} dg$

## Setting

I have a principal $B$-bundle, $Q$, over my space M, with connection $\theta$ (values in the Lie Algebra $\mathfrak{b}$ of the lie group $B$), and a central group extension by $\mathbb{C}^*$,

$$ 0 \to \mathbb{C}^* \to \tilde{B} \to B \to 0$$ where the map $p:\tilde{B} \to B$ is a principal $\mathbb{C}^*$ fibration, yielding an exact sequence of sheaves of groups:

$$ 0 \to \underline{\mathbb{C}}_M^* \to \underline{\tilde{B}}_M \to \underline{B}_M \to 0$$ where $M$ is a paracompact space.

Long story short, I restrict my attention of the space to an open set $ U \subset M$, small enough so that I can "lift" the bundle to a $\tilde{B}$-bundle, $\tilde{Q}$, and the connection to a $\tilde{\mathfrak{b}}$-valued connection $\tilde{\theta}$ on the bundle $\tilde{Q}$. The lifted bundle must satisy the condition that we have a bundle isomorphism $f: \tilde{Q}/\mathbb{C}^* \tilde{\to} Q_{U} $. The lifted connection must satisfy $$q \circ \tilde{\theta} = f^*\theta$$ as 1-forms with values in `$\mathfrak{b} = \tilde{\mathfrak{b}}/\mathbb{C}$; where`

$q$ `is simply the map which quotients out by`

$\mathbb{C}$.

Finally, $g$ is simply a $\mathbb{C}^*$-valued function, which can be thought of as a bundle isomorphism given our local picture is a trivialization. By $g_*$ we simply mean the pullback of $g^{-1}$.

## Warning: Matrices Beware!

I know that this formula is true for vector bundles, or when the group automorphisms are represented by invertible matrices; I have read that literature already. **Unless you are prepared to help me understand why this situation is most certainly in the land of finite vector bundles or matrix groups, such an answer would be redundant.**
The formula also makes sense up to the fact that the transformed connection and original connection must differ by a complex-valued form, which is exactly what this formula prescribes.