ezh

d 1

= ezh

„ (1.17)

1 ’ e’z

dh z

holds for 0 < |z| < 2π. Here the in¬nite order di¬erential operator on the left

is regarded as the limit of the ¬nite order di¬erential operators obtained by

truncating the power series for „ at higher and higher orders.

Let a < b be integers. Then for any non-negative values of h1 and h2 we

have

b+h2 bz az

h2 z e ’h2 z e

zx

’e

e dx = e

z z

a’h1

for z = 0. So if we set

d d

D1 := , D2 := ,

dh1 dh2

1.8. THE UNIVERSAL ENVELOPING ALGEBRA. 19

the for 0 < |z| < 2π we have

b+h2

ebz eaz

’ „ (’z)e’h1 z

ezx dx = „ (z)eh2 z

„ (D1 )„ (D2 )

z z

a’h1

because „ (D1 )f (h2 ) = f (h2 ) when applied to any function of h2 since the con-

stant term in „ is one and all of the di¬erentiations with respect to h1 give

zero.

Setting h1 = h2 = 0 gives

b+h2

eaz ebz

zx

, 0 < |z| < 2π.

„ (D1 )„ (D2 ) e dx = +

1 ’ e’z

1 ’ ez

a’h1

h1 =h2 =0

On then other hand, the geometric sum gives

b

’ e(b’a+1)z

az 1

kz az z 2z (b’a)z

+ ··· + e

e =e 1+e +e =e

1 ’ ez

k=a

eaz ebz

= + .

1 ’ e’z

1 ’ ez

We have thus proved the following exact Euler-MacLaurin formula:

b

b+h2

„ (D1 )„ (D2 ) f (x)dx = f (k), (1.18)

a’h1 k=a

h1 =h2 =0

where the sum on the right is over integer values of k and we have proved this

formula for functions f of the form f (x) = ezx , 0 < |z| < 2π. It is also true

when z = 0 by passing to the limit or by direct evaluation.

Repeatedly di¬erentiating (1.18) (with f (x) = ezx ) with respect to z gives

the corresponding formula with f (x) = xn ezx and hence for all functions of the

form x ’ p(x)ezx where p is a polynomial and |z| < 2π.

There is a corresponding formula with remainder for C k functions.

1.8 The universal enveloping algebra.

We will now give an alternative (algebraic) version of the Campbell-Baker-

Hausdor¬ theorem. It depends on several notions which are extremely important

in their own right, so we pause to develop them.

A universal algebra of a Lie algebra L is a map : L ’ U L where U L is

an associative algebra with unit such that

1. is a Lie algebra homomorphism, i.e. it is linear and

[x, y] = (x) (y) ’ (y) (x)

20 CHAPTER 1. THE CAMPBELL BAKER HAUSDORFF FORMULA

2. If A is any associative algebra with unit and ± : L ’ A is any Lie algebra

homomorphism then there exists a unique homomorphism φ of associative

algebras such that

±=φ—¦ .

It is clear that if U L exists, it is unique up to a unique isomorphism. So

we may then talk of the universal algebra of L. We will call it the universal

enveloping algebra and sometimes put in parenthesis, i.e. write U (L).

In case L = g is the Lie algebra of left invariant vector ¬elds on a group

G, we may think of L as consisting of left invariant ¬rst order homogeneous

di¬erential operators on G. Then we may take U L to consist of all left invariant

di¬erential operators on G. In this case the construction of U L is intuitive

and obvious. The ring of di¬erential operators D on any manifold is ¬ltered by

degree: Dn consisting of those di¬erential operators with total degree at most

n. The quotient, Dn /Dn’1 consists of those homogeneous di¬erential operators

of degree n, i.e. homogeneous polynomials in the vector ¬elds with function

coe¬cients. For the case of left invariant di¬erential operators on a group, these

vector ¬elds may be taken to be left invariant, and the function coe¬cients to be

constant. In other words, (U L)n /(U L)n’1 consists of all symmetric polynomial

expressions, homogeneous of degree n in L. This is the content of the Poincar´- e

Birkho¬-Witt theorem. In the algebraic case we have to do some work to get

all of this. We ¬rst must construct U (L).

1.8.1 Tensor product of vector spaces.

Let E1 , . . . , Em be vector spaces and (f, F ) a multilinear map f : E1 —· · ·—Em ’

F . Similarly (g, G). If is a linear map : F ’ G, and g = —¦ f then we say

that is a morphism of (f, F ) to (g, G). In this way we make the set of all (f, F )

into a category. Want a universal object in this category; that is, an object with

a unique morphism into every other object. So want a pair (t, T ) where T is a

vector space, t : E1 — · · · — Em ’ T is a multilinear map, and for every (f, F )

there is a unique linear map f : T ’ F with

—¦t

f= f

.

Uniqueness. By the universal property t = t —¦t , t = t —¦t so t = ( t —¦ t )—¦t,

but also t = t—¦id. So t —¦ t =id. Similarly the other way. Thus (t, T ), if it

exists, is unique up to a unique morphism. This is a standard argument valid

in any category proving the uniqueness of “initial elements”.

Existence. Let M be the free vector space on the symbols x1 , . . . , xm , xi ∈

Ei . Let N be the subspace generated by all the

(x1 , . . . , xi + xi , . . . , xm ) ’ (x1 , . . . , xi , . . . , xm ) ’ (x1 , . . . , xi , . . . , xm )

and all the

(x1 , . . . , , axi , . . . , xm ) ’ a(x1 , . . . , xi , . . . , xm )

1.8. THE UNIVERSAL ENVELOPING ALGEBRA. 21

for all i = 1, . . . , m, xi , xi ∈ Ei , a ∈ k. Let T = M/N and

t((x1 , . . . , xm )) = (x1 , . . . , xm )/N.

This is universal by its very construction. QED

We introduce the notation