Disclaimer: Nothing in this note is original.

Associated Bundles

Let $P\to M$ be a principal bundle, being principal means that the fiber is the same as the structure group,

\[\text{fiber } F = \text{group} = G.\]

Let $c_ {UV}$ be the transition function

\[U \cap V \to G\]

and let $\rho$ be the representation of $G$, to be specific the $N\times N$ general linear matrix

\[\rho : G\to GL(N)\]

that acts on $\mathbb{C}^{N}$.

We then define a new vector bundle associated to $P$,

\[\pi: E_ {\rho}\to M \text{ or }P_ {\rho}\to M\]

through the representation $\rho$.

The fiber of the associated group is $\mathbb{C}^{N}$. Let $\psi_ {U}$ be the fiber “coordinate” on patch $U$, similarly for patch $V$, we introduce the transition function by making the identification

\[(x,\psi_ {U}) \sim (x,\psi_ {V}) \iff \psi_ {V} = \rho(c_ {VU}(x))\psi_ {U}.\]

Simply put it, we just defined the transition function to be the representation of $G$.

Recall that given a rank-$K$ vector bundle $E\to M$ and local trivialization (which works like a parametrization, instead of coordinate)

\[\Phi_ {U}: F \times M \to \pi ^{-1} U, \quad \mathbb{C}^{n}\times M \to \pi ^{-1} U\]

Then we can introduce a set of basis using $\Phi_ {U}$ by mapping for instance

\[\Phi_ {U}(x,-): (1,0,\dots,0) \mapsto \mathbf{e}_ {U,1},\quad \text{etc.}\]

The transition function is

\[c_ {UV}\in G \subset GL(K).\]

Then we can regard it as the associated vector bundle

\[E\to M := E_ {\rho}\to M\]

of a principal bundle, let’s denote it by $P\to M$. This principal bundle has for fiber $G$, with representation $GL(K)$.

The transition function of $P\to M$ is what? A general element of $P$ is

\[\mathbf{f} = \mathbf{e}_ {U} g_ {U} = \mathbf{e}_ {V} g_ {V},\]

as we know from the property of the vector bundle, the basis transforms as

\[\mathbf{e}_ {V} = \mathbf{e}_ {U} c_ {UV}\]

then we have

\[g_ {U} = c_ {UV} g_ {V}\]

to keep $\mathbf{f}$ basis-independent.

We may say the $P_ {\rho}$ and $E_ {\rho}$ are associated through the representation $\rho$. We give a few examples in the following.

Example 1 The dual bundle. The dual bundle can be regarded as associated to the original bundle through the change of representations

\[g \to g' := (g^{-1} )^{T}.\]