3. The most crucial consequence of the Campbell-Baker-Hausdor¬ formula

is that it shows that the local structure of the Lie group G (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.

10 CHAPTER 1. THE CAMPBELL BAKER HAUSDORFF FORMULA

5. Consider the function „ de¬ned by

w

„ (w) := . (1.3)

1 ’ e’w

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)k bk where

the bk are the Bernoulli numbers.) Then

ψ(z) = „ (log z).

6. The formula is named after three mathematicians, Campbell, Baker, and

Hausdor¬. 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 coe¬cients, 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 di¬erential version of

it:

d

log ((exp A)(exp tB)) = ψ ((exp ad A)(exp t ad B)) B. (1.4)

dt

Since log(exp A(exp tB)) = A when t = 0, integrating (1.4) from 0 to 1 will

prove (1.2). Let us de¬ne “ = “(t) = “(t, A, B) by

“ = log ((exp A)(exp tB)) . (1.5)

Then

exp “ = exp A exp tB

and so

d d

exp “(t) = exp A exp tB

dt dt

= exp A(exp tB)B

= (exp “(t))B so

d

(exp ’“(t)) exp “(t) = B.

dt

We will prove (1.4) by ¬nding a general expression for

d

exp(’C(t)) exp(C(t))

dt

where C = C(t) is a curve in the Lie algebra, g, see (1.11) below.

1.3. THE MAURER-CARTAN EQUATIONS. 11

In our derivation of (1.4) from (1.11) we will make use of an important

property of the adjoint representation which we might as well state now: For

any g ∈ G, de¬ne the linear transformation

Ad g : g ’ g : X ’ gXg ’1 .

(In geometrical terms, this can be thought of as follows: (The di¬erential of )

Left multiplication by g carries g = TI (G) into the tangent space, Tg (G) to G

at the point g. Right multiplication by g ’1 carries this tangent space back to g

and so the combined operation is a linear map of g into itself which we call Ad

g. Notice that Ad is a representation in the sense that

Ad (gh) = (Ad g)(Ad h) ∀g, h ∈ G.

In particular, for any A ∈ g, we have the one parameter family of linear trans-

formations Ad(exp tA) and

d

(exp tA)AX(exp ’tA) + (exp tA)X(’A)(exp ’tA)

Ad (exp tA)X =

dt

(exp tA)[A, X](exp ’tA) so

=

d

Ad(exp tA) —¦ ad A.

Ad exp tA =

dt

But ad A is a linear transformation acting on g and the solution to the di¬er-

ential equation

d

M (t) = M (t)ad A, M (0) = I

dt

(in the space of linear transformations of g) is exp t ad A. Thus Ad(exp tA) =

exp(t ad A). Setting t = 1 gives the important formula

Ad (exp A) = exp(ad A). (1.6)

As an application, consider the “ introduced above. We have

exp(ad “) = Ad (exp “)

= Ad ((exp A)(exp tB))

= (Ad exp A)(Ad exp tB)

= (exp ad A)(exp ad tB)

hence

ad “ = log((exp ad A)(exp ad tB)). (1.7)

1.3 The Maurer-Cartan equations.

If G is a Lie group and γ = γ(t) is a curve on G with γ(0) = A ∈ G, then

A’1 γ is a curve which passes through the identity at t = 0. Hence A’1 γ (0) is

a tangent vector at the identity, i.e. an element of g, the Lie algebra of G.

12 CHAPTER 1. THE CAMPBELL BAKER HAUSDORFF FORMULA

In this way, we have de¬ned a linear di¬erential form θ on G with values in

g. In case G is a subgroup of the group of all invertible n — n matrices (say over

the real numbers), we can write this form as

θ = A’1 dA.

We can then think of the A occurring above as a collection of n2 real valued

functions on G (the matrix entries considered as functions on the group) and

dA as the matrix of di¬erentials of these functions. The above equation giving

θ is then just matrix multiplication. For simplicity, we will work in this case,

although the main theorem, equation (1.8) below, works for any Lie group and

is quite standard.

The de¬nitions of the groups we are considering amount to constraints on

A, and then di¬erentiating these constraints show that A’1 dA takes values in

g, and gives a description of g. It is best to explain this by examples:

• O(n): AA† = I, dAA† + AdA† = 0 or

†

A’1 dA + A’1 dA = 0.

o(n) consists of antisymmetric matrices.

• Sp(n): Let

0 I

J :=

’I 0

and let Sp(n) consist of all matrices satisfying

AJA† = J.

Then

dAJa† + AJdA† = 0

or

(A’1 dA)J + J(A’1 dA)† = 0.

The equation BJ + JB † = 0 de¬nes the Lie algebra sp(n).

• Let J be as above and de¬ne Gl(n,C) to consist of all invertible matrices