Exercise 12, § 4

Lie Groups and Lie Algebras, III, 381

Let $G$ be a finite-dimensional connected real or complex Lie group and $\alpha$ a left invariant differential form of degree $p$ on $G$ . For $\alpha$ to be right invariant too, it is necessary and sufficient that $d\alpha = 0$ . (Observe that the condition for right invariance can be written

\wedge^p(t\mathrm{Ad}(s)) . \alpha(e) = \alpha(e)

for all $s \in G$ and hence it is equivalent to the condition

\sum_{j=1}^f \langle \alpha(e), u_1 \wedge \cdots \wedge u_{j-1} \wedge [u, u_j] \wedge u_{j+1} \wedge \cdots \wedge u_p \rangle = 0

for all $u, u_1, \ldots, u_p$ in $L(G)$ .) In particular, the left and right invariant differential forms of degree 1 on $G$ form a vector space of dimension

\dim L(G) - \dim[L(G), L(G)].

Solution

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