Gauge Transformation of the Vector Potential (on bundles)

Question

Let $D$ be a $G$-Connection on a vector bundle $E$. That is, one can write (locally) any connection $D$ as $D^0 + A$, where $D^0$ is the standard flat connection and $A$ is the vector potential whose components in local coordinates $A_{\mu} \in \text{End}(E)$ live in $\mathfrak{g}$.

Now, let $g \in G$ be a gauge transformation. Under the gauge transformation one can show that the vector potential components transform as $$ A_{\mu}' = gA_{\mu}g^{-1} + g\partial_{\mu}g^{-1}. $$ The claim is that provided $A_{\mu}$ lives in $\mathfrak{g}$, then so will $A'_{\mu}$.

My problem is that it is not obvious to me why this is the case. In particular, the term $$ gA_{\mu}g^{-1}. $$ I'm not even sure how to interpret this term, let alone how it lives in $\mathfrak{g}$. Any thoughts?

[1] Gauge Fields, Knots and Gravity. Baez & Muniain.

Answer 1

If $A\in \mathfrak g\equiv {\rm Lie}[G]$ then $g: A\to gAg^{-1}$ denotes the adjoint representation of $g\in G$ on the algebra

Answer 2

$g X g^{-1}$ denotes the adjoint operation of the group element $g$ on the Lie algebra element $X$. If you wish you can describe this explicitly using the BCH formula. Writing $g=e^Y$ we have $$ e^Y X e^{-Y} = X + [ Y , X ] + \frac{1}{2!} [ Y , [ Y , X ] ] + \frac{1}{3!} [ Y , [ Y , [ Y , X ] ] ] + \cdots $$ The RHS is clearly an element of the Lie algebra. BTW, the second term in the gauge expansion has a similar interpretation $$ e^Y \text{d}e^{-Y} = - d Y + \frac{1}{2!} [ Y ,\text{d}Y] - \frac{1}{3!} [ Y , [ Y , \text{d} Y]] + \cdots $$