1.3. ENVELOPING ALGEBRAS 7 Assertion 1 These operators Xi on S are well-defined. We can show that Xizi 1 · · · zi m is the sum of zizi 1 · · · zi m 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 (Xkzi 1 · · · zi m )−Xk (Xjzi 1 · · · zi m ) = [ Xj, Xk ]zi 1 · · · zi m 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 Xkzjzi 1 · · · zi m 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 zi 2 · · · zi m by zJ. Let us start with the equation Xj (Xkzi 1 · · · zi m ) = Xj (Xkzi 1 zJ) = Xj (Xi 1 (XkzJ) + [Xk, Xi 1 ]zJ) . Note that XkzJ is the sum of zkzJ and a polynomial of degree less than m. If we consider an element XjXi 1 zkzJ, we are in the case i1 j, k and thus we have XjXi 1 zkzJ = Xi 1 XjzkzJ + [ Xj, Xi 1 ]zkzJ. This implies that if we apply XjXi 1 and Xi 1 Xj + [Xj, Xi 1 ] to zkzJ, we have the same element. The same is true if we apply XjXi 1 and Xi 1 Xj + [Xj, Xi 1 ] to a polynomial of degree less than m by the induction hypothesis. Hence, XjXi 1 XkzJ equals Xi 1 XjXkzJ + [Xj, Xi 1 ]XkzJ. We also have that Xj[Xk, Xi 1 ]zJ equals [Xk, Xi 1 ]XjzJ + [Xj, [Xk, Xi 1 ]]zJ by the induction hypothesis. To conclude, Xj (Xkzi 1 · · · zi m ) equals Xi 1 XjXkzJ + [Xj, Xi 1 ]XkzJ + [Xk, Xi 1 ]XjzJ + [Xj, [Xk, Xi 1 ]]zJ. We obtain a similar formula for Xk (Xjzi 1 · · · zi m ). By subtracting this from the above, and using the Jacobi identity, we get Xj (Xkzi 1 · · · zi m ) − Xk (Xjzi 1 · · · zi m ) = Xi 1 [Xj, Xk]zJ + [Xj, [Xk, Xi 1 ]]zJ − [Xk, [Xj, Xi 1 ]]zJ = Xi 1 [Xj, Xk]zJ + [[Xj, Xk], Xi 1 ] zJ. By the induction hypothesis, this equals [Xj, Xk](Xi 1 zJ), which is the same as [Xj, Xk]zi 1 · · · zi m . 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 Xi 1 · · · Xi m (i1 ≤ · · · ≤im) to 1 ∈ S, we get zi 1 · · · zi m . Since these are linearly independent elements, { Xi 1 · · · Xi m | i1 ≤ · · · ≤ im } are linearly independent.

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