Corollary 2

Tag 0ABS, permanent. Cite it as [Bourbaki, Tag 0ABS].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

Suppose that G is finite-dimensional. Let $(e_1, \ldots, e_n)$ be a basis of $L(G)$ , $(e_1^*, \ldots, e_n^*)$ the dual basis, $(c_{ijk})$ the constants of structure of $L(G)$ relative to the basis $(e_1, \ldots, e_n)$ and $\omega_i$ (resp. $\omega'_i$ ) the left (resp. right) invariant differential form on G with values in K such that $(\omega_i)_e = e_i^*$ (resp. $(\omega'_i)_e = e_i^*$ ). Then

d\omega_k + \sum_{i < j} c_{ijk} \omega_i \wedge \omega_j = 0 \quad (k = 1, 2, \ldots, n)

d\omega'_k - \sum_{i < j} c_{ijk} \omega'_i \wedge \omega'_j = 0 \quad (k = 1, 2, \ldots, n).

r < s

(d\omega_k)_e(e_r, e_s) = -(\omega_k)_e([e_r, e_s]) = -\sum_i c_{rsi} (\omega_k)_e(e_i) = -c_{rsk} = -\sum_{i < j} c_{ijk} (\omega_i \wedge \omega_j)_e(e_r, e_s).

The argument is similar for the

\omega'_k

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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Cites

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