Corollary

Tag 066R, permanent. Cite it as [Bourbaki, Tag 066R].

Let $A$ be an associative graded algebra. For two homogeneous elements, $a, b$ of $A$ ,

[\mathrm{ad}_\varepsilon(a), \mathrm{ad}_\varepsilon(b)]_\varepsilon = \mathrm{ad}_\varepsilon([a, b]_\varepsilon).

(25)

It suffices to replace $d$ by $\mathrm{ad}_\varepsilon(a)$ and $\mathrm{ad}_\varepsilon(a)$ by $\mathrm{ad}_\varepsilon(b)$ in (24).

If $\deg a = \alpha, \deg b = \beta$ , formula (25) is equivalent to the following relation for every homogeneous element $c \in A$ of degree $\gamma$

\varepsilon_{\alpha, \gamma}[a, [b, c]_\varepsilon]_\varepsilon + \varepsilon_{\beta, \alpha}[b, [c, a]_\varepsilon]_\varepsilon + \varepsilon_{\gamma, \beta}[c, [a, b]_\varepsilon]_\varepsilon = 0

called the Jacobi identity.

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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