John Stillwell - Naive Lie Theory.pdf - Index of

158 7 The matrix logarithm
Exercises
The congruence relation (6)
F n (A,B) ≡ Lie
− (−1) n F n (B,A)
discovered in the above proof can be strengthened remarkably to
F n (A,B)=−(−1) n F n (B,A).
Here is why.
7.7.1 If Z(A,B) denotes the solution Z of the equation e A e B = e Z , explain why
Z(−B,−A)=−Z(A,B).
7.7.2 Assuming that one may “equate coefficients” for power series in noncommuting
variables, deduce from Exercise 7.7.1 that
F n (A,B)=−(−1) n F n (B,A).
7.8 Discussion
The beautiful self-contained theory of matrix Lie groups seems to have
been discovered by von Neumann [1929]. In this little-known paper 5 von
Neumann defines the matrix Lie groups as closed subgroups of GL(n,C),
and their “tangents” as limits of convergent sequences of matrices. In this
chapter we have recapitulated some of von Neumann’s results, streamlining
them slightly by using now-standard techniques of calculus and linear
algebra. In particular, we have followed von Neumann in using the matrix
exponential and logarithm to move smoothly back and forth between
a matrix Lie group and its tangent space, without appealing to existence
theorems for inverse functions and the solution of differential equations.
The idea of using matrix Lie groups to introduce Lie theory was suggested
by Howe [1983]. The recent texts of Rossmann [2002], Hall [2003],
and Tapp [2005] take up this suggestion, but they move away from the ideas
of von Neumann cited by Howe. All put similar theorems on center stage—
viewing the Lie algebra g of G as both the tangent space and the domain
of the exponential function—but they rely on analytic existence theorems
rather than on von Neumann’s rock-bottom approach through convergent
sequences of matrices.
5 The only book I know that gives due credit to von Neumann’s paper is Godement
[2004], where it is described on p. 69 as “the best possible introduction to Lie groups” and
“the first ‘proper’ exposition of the subject.”