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E n by D_ (ii), - 53 - dj = dkd A (rood Er). Hence by the induction hypothesis, and by w Prop. , that we have unique division by integers). a Q-slgebra, We have thus established (Ar+l). But now (Br+I) follows from (Ar+I) and (Br) by an easy argument. The first equation in Lamina 3 just tells us that ~ is a map of R-algebras. All the maps occuring in the lest two equations of the lemma are homomorphism of algebras preserving identities. In each case it then suffices to verify that the images of the generators di of the algebra E(LF) coincide, and this follows from the explicit description given earlier on.

Then also L = LF -51- Let now conversely F be a given formal group. Since LFC~(Un,PF) , this inclusion map can be pulled back to a homomorphism : E(~) § LD~3 of algebras (universal property of enveloping algebra). n : E(L F) § Un,PF i_~san isomorphism. c; = Cn' ~n" ~ = ~ PROOF of Lemma i Als O Ca @ a ) . D = w n ~. and.. Since R + is divisible and by II w m Prop. 2, the k'~ k form a free basis for Un, and so CL is an isomorphism of modules. Also by D a n d II w Prop. +j=k X k' i+j=k CT ' cai) e CT,CdJ) I i+j=k =k' n z (6k) = ~n CL(dk)~ By extending linearly to E(L), this proves that the first diagram is commutative.

Then : ("Poincar~-Birkhoff - Witt Theorem") L § E(L) is in~ective. We shall accordingly vi~v L as embedded in E(L). % (the order of the factors matters') p_. ee R-module on the dk. d j (k,~ # 0). D(d i) = 1 g d i + d i g l, and hence i+j=k (iv) ~(i) = ~o = i. ~(~) _- O, k#O i, k=O. Now we return to the associative algebra U , p F defined in the preceding section, F being a formal group of dimension n. We -47 shall write ['] - F for the Lie product. Thus [ u,v ]F = PF (u'V) - PF (v'u)" In the notation of Ii w Prop.