Lie-algebra-valued k-form

Let $M$ be a smooth manifold and fix a finite-dimensional real or complex Lie algebra $\mathfrak g$ .

For $k\ge0$ , a $\mathfrak{g}$ -valued differential k-form on $M$ is a smooth section of the vector bundle

\Lambda^k T^*M \otimes \mathfrak{g} \;\longrightarrow\; M.

Here the constant coefficient space means the trivial bundle $M\times\mathfrak g$ ; equivalently, this is a form with values in that bundle.

Concretely, choosing a basis $\{e_a\}$ of $\mathfrak{g}$ , any $\alpha\in\Omega^k(M;\mathfrak{g})$ can be written uniquely as

\alpha = \sum_a \alpha^a \, e_a,

with ordinary $k$ -forms $\alpha^a\in\Omega^k(M)$ .

Equivalent characterizations

Equivalently, it is a smoothly varying alternating multilinear map

\alpha_x : (T_xM)^k \to \mathfrak{g} \quad\text{for each }x\in M.

The space of such forms is commonly denoted $\Omega^k(M;\mathfrak{g})$ .

Examples

Maurer–Cartan form. On a Lie group $G$ , the left Maurer–Cartan form is a $\mathfrak{g}$ -valued 1-form on $G$ .
Connection form. A principal connection on a principal bundle is encoded by a $\mathfrak{g}$ -valued 1-form on the total space (the connection form).
Curvature. The curvature of a principal connection is a $\mathfrak{g}$ -valued 2-form on the total space.