Library / Lie Groups and Lie Algebras / Chapter II / § 1
Proposition 2
Tag 0A1T, permanent. Cite it as [Bourbaki, Tag 0A1T].
Lie Groups and Lie Algebras, chapter II, § 1. Enveloping bigebra of a Lie algebra
We have
(4)
( π u ⊗ π u ) ∘ c = c u + ∘ π u .
(\pi_u \otimes \pi_u) \circ c = c_u^+ \circ \pi_u.
( π u ⊗ π u ) ∘ c = c u + ∘ π u . Let x x x E E E
( π u ∘ π u ) ( c ( x ) ) = ( ( 1 − η u ) ⊗ ( 1 − η u ) ) ( c ( x ) ) = c ( x ) − ( 1 ⊗ η u ) ( c ( x ) ) − ( η u ⊗ 1 ) ( c ( x ) ) + ( η u ⊗ η u ) ( c ( x ) ) .
\begin{align*}
(\pi_u \circ \pi_u)(c(x)) &= ((1 - \eta_u) \otimes (1 - \eta_u))(c(x)) \\
&= c(x) - (1 \otimes \eta_u)(c(x)) - (\eta_u \otimes 1)(c(x)) + (\eta_u \otimes \eta_u)(c(x)).
\end{align*}
( π u ∘ π u ) ( c ( x )) = (( 1 − η u ) ⊗ ( 1 − η u )) ( c ( x )) = c ( x ) − ( 1 ⊗ η u ) ( c ( x )) − ( η u ⊗ 1 ) ( c ( x )) + ( η u ⊗ η u ) ( c ( x )) . As
ε \varepsilon ε is counit of
E E E ,
( 1 ⊗ η u ) ( c ( x ) ) = x ⊗ u , ( η u ⊗ 1 ) ( c ( x ) ) = u ⊗ x
(1 \otimes \eta_u)(c(x)) = x \otimes u, \qquad (\eta_u \otimes 1)(c(x)) = u \otimes x
( 1 ⊗ η u ) ( c ( x )) = x ⊗ u , ( η u ⊗ 1 ) ( c ( x )) = u ⊗ x whence
( η u ⊗ η u ) ( c ( x ) ) = ( η u ⊗ 1 ) ( ( 1 ⊗ η u ) ( c ( x ) ) ) = ε ( x ) . u ⊗ u ;
(\eta_u \otimes \eta_u)(c(x)) = (\eta_u \otimes 1)((1 \otimes \eta_u)(c(x))) = \varepsilon(x).u \otimes u;
( η u ⊗ η u ) ( c ( x )) = ( η u ⊗ 1 ) (( 1 ⊗ η u ) ( c ( x ))) = ε ( x ) . u ⊗ u ; from this we conclude
( π u ⊗ π u ) ( c ( x ) ) = c ( x ) − x ⊗ u − u ⊗ x + ε ( x ) . u ⊗ u .
(\pi_u \otimes \pi_u)(c(x)) = c(x) - x \otimes u - u \otimes x + \varepsilon(x).u \otimes u.
( π u ⊗ π u ) ( c ( x )) = c ( x ) − x ⊗ u − u ⊗ x + ε ( x ) . u ⊗ u . On the other hand,
c u + ( π u ( x ) ) = c ( x ) − x ⊗ u − u ⊗ x + ε ( x ) . u ⊗ u ,
c_u^+(\pi_u(x)) = c(x) - x \otimes u - u \otimes x + \varepsilon(x).u \otimes u,
c u + ( π u ( x )) = c ( x ) − x ⊗ u − u ⊗ x + ε ( x ) . u ⊗ u , whence formula (4).
As E + E^+ E + E E E E + ⊗ E + E^+ \otimes E^+ E + ⊗ E + E ⊗ E E \otimes E E ⊗ E π u ⊗ π u \pi_u \otimes \pi_u π u ⊗ π u E ⊗ E E \otimes E E ⊗ E E + ⊗ E + E^+ \otimes E^+ E + ⊗ E + c u + c_u^+ c u + E + E^+ E + E + ⊗ E + E^+ \otimes E^+ E + ⊗ E + π u \pi_u π u ( E , c ) (E, c) ( E , c ) ( E + , c u + ) (E^+, c_u^+) ( E + , c u + )
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Cites
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Transcribed from the printed volume. Not translated.