Library / Lie Groups and Lie Algebras / Chapter II / § 1
Definition 2
Tag 0A1Z, permanent. Cite it as [Bourbaki, Tag 0A1Z].
Lie Groups and Lie Algebras, chapter II, § 1. Enveloping bigebra of a Lie algebra
Let E E E c c c E E E ( E n ) n ≥ 0 (E_n)_{n \geq 0} ( E n ) n ≥ 0 E E E
E 0 = K .1 , E = ⋃ n ≥ 0 E n E m ⋅ E n ⊂ E m + n for m ≥ 0 , n ≥ 0 c ( E n ) ⊂ ∑ i + j = n Im ( E i ⊗ E j ) for n ≥ 0. †
\begin{align*}
E_0 &= K.1, \quad E = \bigcup_{n \geq 0} E_n \\
E_m \cdot E_n &\subset E_{m+n} \quad \text{for } m \geq 0, n \geq 0 \\
c(E_n) &\subset \sum_{i+j=n} \operatorname{Im}(E_i \otimes E_j) \quad \text{for } n \geq 0.\dagger
\end{align*}
E 0 E m ⋅ E n c ( E n ) = K .1 , E = n ≥ 0 ⋃ E n ⊂ E m + n for m ≥ 0 , n ≥ 0 ⊂ i + j = n ∑ Im ( E i ⊗ E j ) for n ≥ 0. † A bigebra with a filtration compatible with its bigebra structure is called a filtered bigebra.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Transcribed from the printed volume. Not translated.