1.3. ENVELOPING ALGEBRAS 7
Assertion 1 These operators Xi on S are well-defined.
We can show that Xizi1 · · · zim is the sum of zizi1 · · · zim and a polynomial of
degree equal or less than m by induction on m, from which the well-definedness
follows.
Assertion 2 These operators make S into a g-module.
We show Xj (Xkzi1 · · · zim )−Xk (Xjzi1 · · · zim ) = [ Xj, Xk ]zi1 · · · zim by induc-
tion on m. Assume that this holds up to m 1. We first show the formula for the
cases j i1 and k i1. We may assume j k without loss of generality. If j i1,
then the definition of Xkzjzi1 · · · zim implies the formula. If k i1, then we are in
the case j i1 and the formula follows.
Next we show the formula for the case j, k i1. We do not assume j k here.
We abbreviate zi2 · · · zim by zJ . Let us start with the equation
Xj (Xkzi1 · · · zim ) = Xj (Xkzi1 zJ ) = Xj (Xi1 (XkzJ ) + [Xk, Xi1 ]zJ ) .
Note that XkzJ is the sum of zkzJ and a polynomial of degree less than m. If
we consider an element XjXi1 zkzJ , we are in the case i1 j, k and thus we have
XjXi1 zkzJ = Xi1 XjzkzJ + [ Xj, Xi1 ]zkzJ . This implies that if we apply XjXi1
and Xi1 Xj + [Xj, Xi1 ] to zkzJ , we have the same element. The same is true if
we apply XjXi1 and Xi1 Xj + [Xj, Xi1 ] to a polynomial of degree less than m by
the induction hypothesis. Hence, XjXi1 XkzJ equals Xi1 XjXkzJ + [Xj, Xi1 ]XkzJ .
We also have that Xj[Xk, Xi1 ]zJ equals [Xk, Xi1 ]XjzJ + [Xj, [Xk, Xi1 ]]zJ by the
induction hypothesis. To conclude, Xj (Xkzi1 · · · zim ) equals
Xi1 XjXkzJ + [Xj, Xi1 ]XkzJ + [Xk, Xi1 ]XjzJ + [Xj, [Xk, Xi1 ]]zJ .
We obtain a similar formula for Xk (Xjzi1 · · · zim ). By subtracting this from the
above, and using the Jacobi identity, we get
Xj (Xkzi1 · · · zim ) Xk (Xjzi1 · · · zim )
= Xi1 [Xj, Xk]zJ + [Xj, [Xk, Xi1 ]]zJ [Xk, [Xj, Xi1 ]]zJ
= Xi1 [Xj, Xk]zJ + [[Xj, Xk], Xi1 ] zJ .
By the induction hypothesis, this equals [Xj, Xk](Xi1 zJ ), which is the same as
[Xj, Xk]zi1 · · · zim . Hence the result follows.
By Assertion 2 and the universal property of the enveloping algebra, S is a U(g)-
module. Further, if we apply Xi1 · · · Xim (i1 · · · ≤im) to 1 S, we get zi1 · · · zim .
Since these are linearly independent elements, { Xi1 · · · Xim | i1 · · · im } are
linearly independent.
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