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.