01 00 0 1 00

we see that our algebra g with basis A, B and [A, B] = B is indeed the Lie

algebra of the ax + b group.

In a similar way, we could list all possible three dimensional Lie algebras, by

¬rst classifying them according to dim[g, g] and then analyzing the possibilities

for each value of this dimension. Rather than going through all the details, we

list the most important examples of each type. If dim[g, g] = 0 the algebra is

commutative so there is only one possibility.

A very important example arises when dim[g, g] = 1 and that is the Heisen-

berg algebra, with basis P, Q, Z and bracket relations

[P, Q] = Z, [Z, P ] = [Z, Q] = 0.

Up to constants (such as Planck™s constant and i) these are the famous Heisen-

berg commutation relations. Indeed, we can realize this algebra as an algebra

of operators on functions of one variable x: Let P = D = di¬erentiation, let Q

consist of multiplication by x. Since, for any function f = f (x) we have

D(xf ) = f + xf

we see that [P, Q] = id, so setting Z = id, we obtain the Heisenberg algebra.

As an example with dim[g, g] = 2 we have (the complexi¬cation of) the Lie

algebra of the group of Euclidean motions in the plane. Here we can ¬nd a basis

h, x, y of g with brackets given by

[h, x] = y, [h, y] = ’x, [x, y] = 0.

More generally we could start with a commutative two dimensional algebra and

adjoin an element h with ad h acting as an arbitrary linear transformation, A

of our two dimensional space.

The item of study of this chapter is the algebra sl(2) of all two by two

matrices of trace zero, where [g, g] = g.

2.2 sl(2) and its irreducible representations.

Indeed sl(2) is spanned by the matrices:

10 0 1 0 0

h= , e= , f= .

0 ’1 0 0 1 0

They satisfy

[h, e] = 2e, [h, f ] = ’2f, [e, f ] = h.

Thus every element of sl(2) can be expressed as a sum of brackets of elements

of sl(2), in other words

[sl(2), sl(2)] = sl(2).

2.2. SL(2) AND ITS IRREDUCIBLE REPRESENTATIONS. 37

The bracket relations above are also satis¬ed by the matrices

« « «

20 0 02 0 0 00

ρ2 (h) := 0 0 0 , ρ2 (e) := 0 0 1 , ρ2 (f ) := 1 0 0 ,

’2

00 00 0 0 20

the matrices

« « «

3 00 0 0 3 0 0 0 0 0 0

¬0 10 0· ¬0 0 2 0· ¬1 0 0 0·

· ·

ρ3 (h) := ¬ , ρ (e) := ¬ , ρ (f ) := ¬ ·,

0 ’1 0 3 1 3

0 0 0 0 0 2 0 0

0 0 ’3

0 0 0 0 0 0 0 3 0

the (n + 1) — (n + 1) matrices given

and, more generally, by

« «

··· ··· ··· ···

n 0 0 0 n 0

¬0 n ’ 2 ··· ··· n ’ 1 ···

0· ¬0 0 0·

¬ · ¬ ·

ρn (h) := ¬ . . . . . · , ρ (e) = ¬ . . . . .· ,

¬. . . . .· n ¬. . . . .·

. . . . .· ¬. . . . .·

¬

· · · ’n + 2 0 ··· ···

0 0 0 0 1

··· ··· ’n ··· ···

0 0 0 0 0

«

0 ··· ···

0 0

···

¬1 0 0·

ρn (f ) := ¬ . .· .

¬ ·

. . .

. . . . .

. . . . .

0 ···

0 n 0

These representations of sl(2) are all irreducible, as is seen by successively

applying ρn (e) to any non-zero vector until a vector with non-zero element in

the ¬rst position and all other entries zero is obtained. Then keep applying

ρn (f ) to ¬ll up the entire space.

These are all the ¬nite dimensional irreducible representations of sl(2) as

can be seen as follows: In U (sl(2)) we have

[h, f k ] = ’2kf k , [h, ek ] = 2kek (2.1)

[e, f k ] = ’k(k ’ 1)f k’1 + kf k’1 h. (2.2)

Equation (2.1) follows from the fact that bracketing by any element is a deriva-

tion and the fundamental relations in sl(2). Equation (2.2) is proved by induc-

tion: For k = 1 it is true from the de¬ning relations of sl(2). Assuming it for

k, we have

[e, f k+1 ] [e, f ]f k + f [e, f k ]

=

hf k ’ k(k ’ 1)f k + kf k h

=

[h, f k ] + f k h ’ k(k ’ 1)f k + kf k h

=

’2kf k ’ k(k ’ 1)f k + (k + 1)f k h

=

’(k + 1)kf k + (k + 1)f k h.

=

38 CHAPTER 2. SL(2) AND ITS REPRESENTATIONS.

We may rewrite (2.2) as

1k 1 1

f k’1 + f k’1 h.

e, f = (’k + 1) (2.3)

(k ’ 1)! (k ’ 1)!

k!

In any ¬nite dimensional module V , the element h has at least one eigenvector.

This follows from the fundamental theorem of algebra which assert that any

polynomial has at least one root; in particular the characteristic polynomial of

any linear transformation on a ¬nite dimensional space has a root. So there is

a vector w such that hw = µw for some complex number µ. Then

h(ew) = [h, e]w + ehw = 2ew + µew = (µ + 2)(ew).

Thus ew is again an eigenvector of h, this time with eigenvalue µ+2. Successively

applying e yields a vector v» such that

hv» = »v» , ev» = 0. (2.4)

Then U (sl(2))v» is an invariant subspace, hence all of V . We say that v is a

cyclic vector for the action of g on V if U (g)v = V ,

We are thus led to study all modules for sl(2) with a cyclic vector v» satis-

fying (2.4). In any such space the elements

1k