12 A2B+AB2+B2A+BA2−2ABA−2BAB]
= 1
12[A,[A, B]]+1
12[B,[B, A]].
This suggests that the series for log[(expA)(expB)] can be expressed entirely in terms of successive Lie brackets ofAandB. This is so, and is the content of the Campbell-Baker-Hausdorff formula.
One of the important consequences of the mere existence of this formula is the following. Suppose thatgis the Lie algebra of a Lie groupG. Then thelocal structure ofGnear the identity, i.e. the rule for the product of two elements of Gsufficiently closed to the identity is determined by its Lie algebrag. Indeed, the exponential map is locally a diffeomorphism from a neighborhood of the origin ingonto a neighborhoodW of the identity, and ifU ⊂W is a (possibly smaller) neighborhood of the identity such thatU·U ⊂W, the the product of a= expξ andb= expη, with a∈U andb∈U is then completely expressed in terms of successive Lie brackets ofξandη.
We will give two proofs of this important theorem. One will be geometric -the explicit formula for -the series for log[(expA)(expB)] will involve integration, and so makes sense over the real or complex numbers. We will derive the formula from the “Maurer-Cartan equations” which we will explain in the course of our discussion. Our second version will be more algebraic. It will involve such ideas as the universal enveloping algebra, comultiplication and the Poincar´ e-Birkhoff-Witt theorem. In both proofs, many of the key ideas are at least as important as the theorem itself.
1.2 The geometric version of the CBH formula.
To state this formula we introduce some notation. Let adAdenote the operation of bracketing on the left byA, so
adA(B) := [A, B].
Define the functionψby
ψ(z) = zlogz z−1
which is defined as a convergent power series around the pointz= 1 so ψ(1 +u) = (1 +u)log(1 +u)
1.2. THE GEOMETRIC VERSION OF THE CBH FORMULA. 9 In fact, we will also take this as adefinition of the formal power series forψin terms ofu. The Campbell-Baker-Hausdorff formula says that
log((expA)(expB)) =A+ Z 1
0
ψ((exp adA)(exptadB))Bdt. (1.2) Remarks.
1. The formula says that we are to substitute u= (exp adA)(exptadB)−1
into the definition ofψ, apply this operator to the elementBand then integrate.
In carrying out this computation we can ignore all terms in the expansion of ψ in terms of ad A and ad B where a factor of ad B occurs on the right, since (adB)B = 0. For example, to obtain the expansion through terms of degree three in the Campbell-Baker-Hausdorff formula, we need only retain quadratic and lower order terms in u, and so
u = adA+1 where the dots indicate either higher order terms or terms with adB occurring on the right. So up through degree three (1.2) gives
log(expA)(expB) =A+B+1
2[A, B] + 1
12[A,[A, B]]− 1
12[B,[A, B]] +· · · agreeing with our preceding computation.
2. The meaning of the exponential function on the left hand side of the Campbell-Baker-Hausdorff formula differs from its meaning on the right. On the right hand side, exponentiation takes place in the algebra of endomorphisms of the ring in question. In fact, we will want to make a fundamental reinter-pretation of the formula. We want to think of A, B, etc. as elements of a Lie algebra, g. Then the exponentiations on the right hand side of (1.2) are still taking place in End(g). On the other hand, ifgis the Lie algebra of a Lie group G, then there is an exponential map: exp: g→G, and this is what is meant by the exponentials on the left of (1.2). This exponential map is a diffeomorphism on some neighborhood of the origin ing, and its inverse, log, is defined in some neighborhood of the identity in G. This is the meaning we will attach to the logarithm occurring on the left in (1.2).
3. The most crucial consequence of the Campbell-Baker-Hausdorff formula is that it shows that the local structure of the Lie groupG(the multiplication law for elements near the identity) is completely determined by its Lie algebra.
4. For example, we see from the right hand side of (1.2) that group multi-plication and group inverse are analytic if we use exponential coordinates.
5. Consider the functionτ defined by τ(w) := w
1−e−w. (1.3)
This is a familiar function from analysis, as it enters into the Euler-Maclaurin formula, see below. (It is the exponential generating function of (−1)kbk where thebk are the Bernoulli numbers.) Then
ψ(z) =τ(logz).
6. The formula is named after three mathematicians, Campbell, Baker, and Hausdorff. But this is a misnomer. Substantially earlier than the works of any of these three, there appeared a paper by Friedrich Schur, “Neue Begruendung der Theorie der endlichen Transformationsgruppen,” Mathematische Annalen 35 (1890), 161-197. Schur writes down, as convergent power series, the com-position law for a Lie group in terms of ”canonical coordinates”, i.e., in terms of linear coordinates on the Lie algebra. He writes down recursive relations for the coefficients, obtaining a version of the formulas we will give below. I am indebted to Prof. Schmid for this reference.
Our strategy for the proof of (1.2) will be to prove a differential version of it:
d
dtlog ((expA)(exptB)) =ψ((exp adA)(exptadB))B. (1.4) Since log(expA(exptB)) = A when t = 0, integrating (1.4) from 0 to 1 will prove (1.2). Let us define Γ = Γ(t) = Γ(t, A, B) by
Γ = log ((expA)(exptB)). (1.5)
Then
exp Γ = expAexptB and so
d
dtexp Γ(t) = expAd dtexptB
= expA(exptB)B
= (exp Γ(t))B so (exp−Γ(t))d
dtexp Γ(t) = B.
We will prove (1.4) by finding a general expression for exp(−C(t))d
dtexp(C(t))
whereC=C(t) is a curve in the Lie algebra,g, see (1.11) below.
1.3. THE MAURER-CARTAN EQUATIONS. 11