---
date: 2022-02-11 23:24
modification date: Friday 11th February 2022 23:24:55
title: Lie algebra valued form
aliases: [Lie algebra valued form, adjoint bundle, bundle valued form]

---

Tags: 
?
Refs:
?

# Bundle valued forms

For $E\searrow M$ a [vector bundle](Unsorted/vector%20bundles.md), an **$E\dash$valued differential form** on $M$ is
$$
\drcomplex_{M}(E) \da \globsec{E\tensor \Extcomplex \T\dual M} \cong \drcomplex_M \tensor_{C^\infty(M; \RR)} \globsec{E}
$$

# Lie algebra valued forms
$$
\cocomplex{\Omega}_{X}(M, \lieg) \da \globsec{( \lieg \times M) \tensor \Extcomplex \T\dual M}
$$
# The adjoint bundle

Commonly used to define the adjoint bundle: for $P$ a [principal bundle](Unsorted/principal%20bundle.md) with structure group $G\in \Lie\Grp$ and $\Lie(G) \da \lieg \in \Lie\Alg$, since there is a big adjoint action
$$
G\actson \lieg: \Ad_g(M) = gMg\inv
$$
yielding a representation of $G$. Thus one can form the [associated bundle](Unsorted/associated%20bundle.md) 
$$
\ad P \da P \mix{\Ad}\lieg,\qquad (pg, x)\sim (p, \Ad_g(x))
$$

![](attachments/Pasted%20image%2020220502175219.png)