A crash course on Kleinian groups; lectures given at a by American Mathematical Society

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A(e) : Un,PF § (c) is an isomorphism of bialgebras. by ~(e) o ~F = ~G ~ E(e). Define -55- l(e) . 2F o ~F = Thus (cf. (a) with W = ICe)), x(e) : e. On the other hand, for given ~ ~ nG o JG o e. V , we have by (a) and (b) ~ ~F = 2G ~ E(uCv)) ~ ~F' hence and so by (c) (with e = ~(~)), = Thus A and ~ are inverse bijections. established. The theorem has thus been (Incidentally we have proved also that E is a bij ection. en every commutative formal group F is isomorphic t_~oth__~eadditive ~ of dimension dim F.

If R + is torsion free then Lf = 0 if and onl~ if f - O. By I, w Theorem 2. For the rest of this section we assume that R is a Q-algebra (Q is the field of rational numbers). Under this hypothesis we shall prove that the category of formal groups and the category of Lie algebras which are free E-modules of finite rank are isomorphic. More precisely we have: THE O R ~ i (i) Let R b e a Q - ~ . For each Lie_ algebra L which is a free module of finite dimension over R, there exists a formal ~rou~ F such that L i s isomorphic to 2 " (ii) (iii) on ~ Hom~(F,G) § HOmLie(~,L G) is abijection.

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. Similarly for the second diagram. ) qT, - ~, Un -52- is comnutative. That qL defines a bialgebra structure on U n is now trivial by Lemma i. The isomorphism of categories ~ ~ of the last section ensures the existence of a formal group F such that t h e Un,qL = Un,PF. Since CL maps d i onto Ai, L and are isomorphic under CL as modules, and since CL preserves the Lie product then this is an isomorphism of Lie algebras.

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