John Stillwell - Naive Lie Theory.pdf - Index of

7.8 Discussion 159
Indeed, von Neumann’s purpose in pursuing elementary constructions
in Lie theory was to explain why continuity apparently implies differentiability
for groups, a question raised by Hilbert in 1900 that became known as
Hilbert’s fifth problem. It would take us too far afield to explain Hilbert’s
fifth problem more precisely than we have already done in Section 7.3,
other than to say that von Neumann showed that the answer is yes for compact
groups, and that Gleason, Montgomery, and Zippin showed in 1952
that the answer is yes for all groups.
As mentioned in Section 4.7, Hamilton made the first extension of the
exponential function to a noncommutative domain by defining it for quaternions
in 1843. He observed almost immediately that it maps the pure imaginary
quaternions onto the unit quaternions, and that e q+q′ = e q e q′ when
qq ′ = q ′ q. He took the idea further in his Elements of Quaternions of
1866, realizing that e q+q′ is not usually equal to e q e q′ , because of the noncommutative
quaternion product. On p. 425 of Volume I he actually finds
the second-order approximation to the Campbell–Baker–Hausdorff series:
e q+q′ − e q e q′ = qq′ − q ′ q
2
+ terms of third and higher dimensions.
The early proofs (or attempted proofs) of the general Campbell–Baker–
Hausdorff theorem around 1900 were extremely lengthy—around 20 pages.
The situation did not improve when Bourbaki developed a more conceptual
approach to the theorem in the 1960s. See for example Serre [1965], or
Section 4 or Bourbaki [1972], Chapter II. Bourbaki believes that the proper
setting for the theorem is in the framework of free magmas, free algebras,
free groups, and free Lie algebras, all of which takes longer to explain
than the proofs by Campbell, Baker, and Hausdorff. It seems to me that
these proofs are totally outclassed by the Eichler proof I have used in this
chapter, which assumes only that the variables A, B, C have an associative
product, and uses only calculations that a high-school student can follow.
Martin Eichler (1912–1992) was a German mathematician (later living
in Switzerland) who worked mainly in number theory and related parts of
algebra and analysis. A famous saying, attributed to him, is that there are
five fundamental operations of arithmetic: addition, subtraction, multiplication,
division, and modular forms. Some of his work involves orthogonal
groups, but nevertheless his 1968 paper on the Campbell–Baker–Hausdorff
theorem seems to come out of the blue. Perhaps this is a case in which an
outsider saw the essence of a theorem more clearly than the experts.