John Stillwell - Naive Lie Theory.pdf - Index of

7.7 Eichler’s proof of Campbell–Baker–Hausdorff 157
Relations (6) and (7) allow us to pass from polynomials in A, B to
polynomials in A/2, B/2, paving the way for another application of fact 3
and a new relation, between F n (A,B) and itself.
Relation (6) allows us to rewrite the two terms on the right side of (7)
as follows:
F n (A/2,B) ≡ Lie F n (A/2,B/2) − F n (A/2 + B,−B/2) by (6)
≡ Lie F n (A/2,B/2)+F n (A/2 + B/2,B/2) by (3)
≡ Lie 2 −n F n (A,B)+2 −n F n (A + B,B) by fact 3,
F n (−A/2,A + B)
≡ Lie F n (−A/2,A/2+B/2)−F n (A/2+B,−A/2−B/2) by (6)
≡ Lie − F n (A/2,B/2)+F n (B/2,A/2 + B/2) by (4) and (3)
≡ Lie − 2 −n F n (A,B)+2 −n F n (B,A + B) by fact 3.
So (7) becomes
F n (A,B) ≡ Lie 2 1−n F n (A,B)+2 −n F n (A + B,B) − 2 −n F n (B,A + B),
and, with the help of (5), this simplifies to
(1 − 2 1−n )F n (A,B) ≡ Lie 2 −n (1 +(−1) n )F n (A + B,B). (8)
If n is odd, (8) already shows that F n (A,B) ≡ Lie 0.
If n is even, we replace A by A − B in (8), obtaining
Theleftsideof(9)
(1 − 2 1−n )F n (A − B,B) ≡ Lie 2 1−n F n (A,B). (9)
(1 − 2 1−n )F n (A − B,B) ≡ Lie − (1 − 2 1−n )F n (A,−B) by (3),
so, making this replacement, (9) becomes
2 1−n
− F n (A,−B) ≡ Lie
1 − 2 1−n F n(A,B). (10)
Finally, replacing B by −B in (10), we get
2 1−n
−F n (A,B) ≡ Lie
1 − 2 1−n F n(A,−B)
( ) 2
1−n 2
≡ Lie −
1 − 2 1−n F n (A,B) by (10),
and this implies F n (A,B) ≡ Lie 0, as required.
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