John Stillwell - Naive Lie Theory.pdf - Index of

154 7 The matrix logarithm
The idea of representing the Z in e Z = e X e Y by a power series in noncommuting
variables X and Y allows us to prove the converse of the theorem that XY = YX
implies e X e Y = e X+Y .
7.6.3 Suppose that e X e Y = e Y e X . By appeal to the proof of the log multiplicative
property in Section 7.1, or otherwise, show that XY = YX.
7.6.4 Deduce from Exercise 7.6.3 that e X e Y = e X+Y if and only if XY = YX.
7.7 Eichler’s proof of Campbell–Baker–Hausdorff
To facilitate an inductive proof, we let
e A e B = e Z , Z = F 1 (A,B)+F 2 (A,B)+F 3 (A,B)+···, (*)
where F n (A,B) is the sum of all the terms of degree n in Z, and hence is a
homogeneous polynomial of degree n in the variables A and B. Since the
variables stand for matrices in the Lie algebra g, they do not generally commute,
but their product is associative. From the calculation in the previous
section we have
F 1 (A,B)=A + B,
F 2 (A,B)= 1 2 (AB − BA)= 1 2 [A,B].
We will call a polynomial p(A,B,C,...) Lie if it is a linear combination
of A,B,C,... and (possibly nested) Lie bracket terms in A,B,C, .... Thus
F 1 (A,B) and F 2 (A,B) are Lie polynomials, and the theorem we wish to
prove is:
Campbell–Baker–Hausdorff theorem. For each n ≥ 1, the polynomial
F n (A,B) in (*) is Lie.
Proof. Since products of A,B,C,... are associative, the same is true of
products of power series in A,B,C,...,soforanyA,B,C we have
∞
∑
i=1F i
( ∞∑
j=1
(e A e B )e C = e A (e B e C ),
and therefore, if e A e B e C = e W ,
)
W = F j (A,B),C =
∞
∑
i=1
F i
(A,
∞
∑
j=1
)
F j (B,C) . (1)
Our induction hypothesis is that F m is a Lie polynomial for m < n, andwe
wish to prove that F n is Lie.