I was glad to see that Wikipedia’s page for the Baker-Campbell-Hausdorff formula actually explicitly states the formula. When I was first learning the subject of Lie groups and algebras, the authors would only show the first few terms, and then an ellipsis. It always left me with the impression that the actual formula was so hideous that no one ever mentioned it explicitly, but only passed over it in discreet silence.
Ars Mathematica
Dedicated to the mathematical arts.
“… There is no expression in closed form for an arbitrary Lie algebra, though there are exceptional tractable cases, as well as efficient algorithms for working out the expansion in applications….”
From your point of view, what is a particularly pretty exceptional tractable cases, or how efficient is an “efficient algorithm”, or what sequences of coefficients are surprisingly from some other area of Mathematics. Please?
Have you read Godement’s wonderful book on the subject? The part on Campbell-Hausdorff’s formula is particularily funny :
“Even if some authors take it seriously, Campbell-Hausdorff’s formula only presents a limited interest for advanced Lie group theory and one experimentally remarks that it is possible to read or write thousands of pages on Lie groups, for instance on their (finite and infinite-dimensional) representations, without having to use it : the main interest of this result is pedagogical and to bring some simple proofs of the theorems which form the rest of this section. Moreover, the authors that cover this subject are careful not to use the formula, and if they seldom show the first terms :
H(X,Y) = X + Y + 1/2 [X,Y] + 1/12 [X,[X,Y]] + 1/12 [Y,[Y,X]] – 1/24 [X,[Y,[X,Y]]] + …, (6.12)
it is impossible to find on the market anyone who shows also the proof of 6.12. Seemingly, someone, probably at the beginning of the century, was charitable enough to compute precisely the formula and to become anonymously famous, through very convenient “remarks” or “exercises”.
One easily understands their discretion on this topic by trying to prove (6.12) from (6.8).
[two pages of infinitely boring computations]
Exercise. Imagine an exercise which perfectly fits here, and give the solution”.
[Sorry for my terrible translation]
Jonathan: The only case I know is one where all of the brackets past a certain point are zero; then the sum is finite. Do you have a particular question in mind?
Irakos: No, I haven’t. ( I checked my local library, and they don’t have it, either.) It’s interesting that he made the same observation that you rarely see the actual formula. It’s been a long time since I learned Lie theory, so I don’t even remember what it’s used for.