In AX let I be the two-sided ideal generated by all elements of the form aa, a ∈

AX and (ab)c + (bc)a + (ca)b, a, b, c ∈ AX . We set

LX := AX /I

and call LX the free Lie algebra on X. Any map from X to a Lie algebra L

extends to a unique algebra homomorphism from LX to L.

We claim that the ideal I de¬ning LX is graded. This means that if a = an

is a decomposition of an element of I into its homogeneous components, then

each of the an also belong to I. To prove this, let J ‚ I denote the set of all

a= an with the property that all the homogeneous components an belong

to I. Clearly J is a two sided ideal. We must show that I ‚ J. For this it is

enough to prove the corresponding fact for the generating elements. Clearly if

a= ap , b = bq , c = cr

then

(ab)c + (bc)a + (ca)b = ((ap bq )cr + (bq cr )ap + (cr ap )bq ) .

p,q,r

But also if x = xm then

x2 = x2 + (xm xn + xn xm )

n

m<n

and

xm xn + xn xm = (xm + xn )2 ’ x2 ’ x2 ∈ I

m n

so I ‚ J.

The fact that I is graded means that LX inherits the structure of a graded

algebra.

1.11. FREE LIE ALGEBRAS 31

1.11.3 The free associative algebra Ass(X).

Let VX be the vector space of all ¬nite formal linear combinations of elements

of X. De¬ne

AssX = T (VX ),

the tensor algebra of VX . Any map of X into an associative algebra A extends to

a unique linear map from VX to A and hence to a unique algebra homomorphism

from AssX to A. So AssX is the free associative algebra on X.

We have the maps X ’ LX and : LX ’ U (LX ) and hence their com-

position maps X to the associative algebra U (LX ) and so extends to a unique

homomorphism

Ψ : AssX ’ U (LX ).

On the other hand, the commutator bracket gives a Lie algebra structure to

AssX and the map X ’ AssX thus give rise to a Lie algebra homomorphism

LX ’ AssX

which determines an associative algebra homomorphism

¦ : U (LX ) ’ AssX .

both compositions ¦ —¦ Ψ and Ψ —¦ ¦ are the identity on X and hence, by unique-

ness, the identity everywhere. We obtain the important result that U (LX ) and

AssX are canonically isomorphic:

U (LX ) ∼ AssX . (1.23)

=

Now the Poincar´-Birkho¬ -Witt theorem guarantees that the map : LX ’

e

U (LX ) is injective. So under the above isomorphism, the map LX ’ AssX is

injective. On the other hand, by construction, the map X ’ VX induces a

surjective Lie algebra homomorphism from LX into the Lie subalgebra of AssX

generated by X. So we see that the under the isomorphism (1.23) LX ‚ U (LX )

is mapped isomorphically onto the Lie subalgebra of AssX generated by X.

Now the map

X ’ AssX — AssX , x’x—1+1—x

extends to a unique algebra homomorphism

∆ : AssX ’ AssX — AssX .

Under the identi¬cation (1.23) this is none other than the map

∆ : U (LX ) ’ U (LX ) — U (LX )

and hence we conclude that LX is the set of primitive elements of AssX :

LX = {w ∈ AssX |∆(w) = w — 1 + 1 — w.} (1.24)

under the identi¬cation (1.23).

32 CHAPTER 1. THE CAMPBELL BAKER HAUSDORFF FORMULA

1.12 Algebraic proof of CBH and explicit for-

mulas.

We recall our constructs of the past few sections: X denotes a set, LX the free

Lie algebra on X and AssX the free associative algebra on X so that AssX

may be identi¬ed with the universal enveloping algebra of LX . Since AssX may

be identi¬ed with the non-commutative polynomials indexed by X, we may

consider its completion, FX , the algebra of formal power series indexed by X.

Since the free Lie algebra LX is graded we may also consider its completion

which we shall denote by LX . Finally let m denote the ideal in FX generated

by X. The maps

exp : m ’ 1 + m, log : 1 + m ’ m

are well de¬ned by their formal power series and are mutual inverses. (There is

no convergence issue since everything is within the realm of formal power series.)

Furthermore exp is a bijection of the set of ± ∈ m satisfying ∆± = ± — 1 + 1 — ±

to the set of all β ∈ 1 + m satisfying ∆β = β — β.

1.12.1 Abstract version of CBH and its algebraic proof.

In particular, since the set {β ∈ 1 + m|∆β = β — β} forms a group, we conclude

that for any A, B ∈ LX there exists a C ∈ LX such that

exp C = (exp A)(exp B).

This is the abstract version of the Campbell-Baker-Hausdor¬ formula. It de-

pends basically on two algebraic facts: That the universal enveloping algebra of

the free Lie algebra is the free associative algebra, and that the set of primitive

elements in the universal enveloping algebra (those satisfying ∆± = ±—1+1—±)

is precisely the original Lie algebra.

1.12.2 Explicit formula for CBH.

De¬ne the map

¦ : m © AssX ’ LX ,

¦(x1 . . . xn ) := [x1 , [x2 , . . . , [xn’1 , xn ] · · · ] = ad(x1 ) · · · ad(xn’1 )(xn ),

and let ˜ : AssX ’ End(LX ) be the algebra homomorphism extending the Lie

algebra homomorphism ad : LX ’ End(LX ). We claim that

¦(uv) = ˜(u)¦(v), ∀ u ∈ AssX , v ∈ m © AssX . (1.25)

Proof. It is enough to prove this formula when u is a monomial, u = x1 · · · xn .

We do this by induction on n. For n = 0 the assertion is obvious and for n = 1

1.12. ALGEBRAIC PROOF OF CBH AND EXPLICIT FORMULAS. 33

it follows from the de¬nition of ¦. Suppose n > 1. Then

¦(x1 · · · xn v) = ˜(x1 )¦(x2 · · · xn v)

= ˜(x1 )˜(x2 . . . xn )¦(v)

= ˜(x1 · · · xn )¦(v). QED

Let Ln denote the n’th graded component of LX . So L1 consists of linear

X X

2

combinations of elements of X, LX is spanned by all brackets of pairs of elements

of X, and in generalLn is spanned by elements of the form

X