John Stillwell - Naive Lie Theory.pdf - Index of

7.6 The Campbell–Baker–Hausdorff theorem 153
that
Z = log(e X e Y )=
(X +Y + XY + X 2
2! + Y 2
···)
2! +
− 1 (X +Y + XY + X 2
2
2! + Y 2
···) 2
2! +
+ 1 (X +Y + XY + X 2
3
2! + Y 2
···) 3
2! +
−···
= X +Y + 1 2 XY − 1 YX+ higher-order terms
2
= X +Y + 1 [X,Y ]+higher-order terms.
2
The hard part of the Campbell-Baker-Hausdorff theorem is to prove that
all the higher-order terms are composed from X and Y by Lie brackets.
Campbell attempted to do this in 1897. His work was amended by
Baker in 1905, with further corrections by Hausdorff producing a complete
proof in 1906. However, these first proofs were very long, and many
attempts have since been made to derive the theorem with greater economy
and insight. Modern textbook proofs are typically only a few pages long,
but they draw on differentiation, integration, and specialized machinery
from Lie theory.
The most economical proof I know is one by Eichler [1968]. It is only
two pages long and purely algebraic, showing by induction on n that all
terms of order n > 1 are linear combinations of Lie brackets. The algebra
is very simple, but ingenious (as you would expect, since the theorem is
surely not trivial). In my opinion, this is also an insightful proof, showing
as it does that the theorem depends only on simple algebraic facts. I present
Eichler’s proof, with some added explanation, in the next section.
Exercises
7.6.1 Show that the cubic term in log(e X e Y ) is
1
12 (X 2 Y + XY 2 +YX 2 +Y 2 X − 2XYX − 2YXY).
7.6.2 Show that the cubic polynomial in Exercise 7.6.1 is a linear combination of
[X,[X,Y]] and [Y,[Y,X]].