Bi-invariant differential form - Imported Knowlpedia Content

Let $G$ be a Lie group. For $g\in G$ , write $L_g$ and $R_g$ for left translation and right translation.

Definition. A differential $k$ -form $\omega\in \Omega^k(G)$ is bi-invariant if

L_g^*\omega=\omega \quad\text{and}\quad R_g^*\omega=\omega \qquad \text{for all } g\in G.

Equivalent characterizations

Equivalently, $\omega$ is both left-invariant and right-invariant.

Characterization (connected case). If $G$ is connected, a left-invariant form is determined by its value at the identity $e$ , i.e. by an alternating multilinear map $\omega_e:\wedge^k\mathfrak{g}\to \Bbbk$ . Such a form is bi-invariant if and only if $\omega_e$ is invariant under the adjoint action:

\omega_e(\mathrm{Ad}_g X_1,\dots,\mathrm{Ad}_g X_k)=\omega_e(X_1,\dots,X_k)\quad\text{for all }g\in G.

Remarks

Motivation. Bi-invariant forms capture intrinsic geometry on $G$ compatible with both left and right symmetries. For example, a bi-invariant metric determines a bi-invariant volume form, and Ad-invariant forms on $\mathfrak{g}$ are the starting point for Chern–Weil constructions on homogeneous spaces.