By John Edward Campbell
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Tr ad Ft / ı Ft ! X; Y /, our claim easily follows. 3 Non-uniqueness The Lie series F and G are not uniquely determined by (KV1) (for instance one may add X to F and Y to G), not even by the system (KV1)(KV2) as shown by the next elementary proposition. In Sect. 7 we shall also give two ways of constructing new solutions of (KV1)(KV2) from a given one. We need a preliminary lemma. 7. Let U be an open subset of g g and F W U ! g X; g Y / 2 U, g 2 G. X; Y /; as endomorphisms of g. Proof. g X; g Y / 2 U for g D exp tZ and jtj small enough.
X1 ; x2 / WD '. x1 / '. x1 x2 / x1 '. tY; X / D X C t'. 6 in the arXiv version of ; cf. 34 below). u; v/ 42 1 Kashiwara-Vergne Method for Lie groups where c is a symmetric series. X / D c. X; X / is an even series. (e) Let us gather all pieces. 14. 1 2 X 2 coth X2 1 . 14. This will prove the Kashiwara-Vergne conjecture in those cases. Working first with l2 as before, let B denote the algebra of all (associative but non-commutative) formal series in x D ad X and y D ad Y . l2 / extends to a morphism of associative algebras ad W A2 !
Y; Z// satisfied by the Campbell-Hausdorff law, a consequence of the associativity of the group law and the definition of V . To exploit it we now work in the (completed) free Lie algebra l3 with generators X; Y; Z. X C Y / C Z where F1;2 2 TA3 means F acting on the first and second generators of l3 , trivially on the third, and F12;3 2 TA3 means F acting on X C Y and Z. F; G/, will eventually lead to the conclusion. 4 A “Divergence” and a “Jacobian” In An , the completion of the free associative algebra with generators X1 ; ::; Xn over K, we denote by a D a0 C n X a k Xk kD1 the unique decomposition of any a 2 An with a0 2 K and ak 2 An .