Quantum Mechanics for Mathematicians

of the map at the identity. For general Lie groupsG, something similar can
be done, showing that a representationπofGgives a representation of the Lie
algebra (by taking the derivative at the identity), and then trying to classify
Lie algebra representations.

Theorem.Ifπ:G→GL(n,C)is a group homomorphism, then

π′:X∈g→π′(X) =

d

dt

(π(etX))|t=0∈gl(n,C) =M(n,C)

satisfies

1.

π(etX) =etπ

′(X)

2. Forg∈G
   π′(gXg−^1 ) =π(g)π′(X)(π(g))−^1

3.π′is a Lie algebra homomorphism:

π′([X,Y]) = [π′(X),π′(Y)]

Proof. 1. We have

d

dt

π(etX) =

d

ds

π(e(t+s)X)|s=0

=

d

ds

π(etXesX)|s=0

=π(etX)

d

ds

π(esX)|s=0

=π(etX)π′(X)

Sof(t) =π(etX) satisfies the differential equationdtdf =fπ′(X) with

initial conditionf(0) = 1. This has the unique solutionf(t) =etπ

′(X)

2. We have

etπ

′(gXg− (^1) )
=π(etgXg
− 1
)
=π(getXg−^1 )
=π(g)π(etX)π(g)−^1
=π(g)etπ
′(X)
π(g)−^1
Differentiating with respect totatt= 0 gives
π′(gXg−^1 ) =π(g)π′(X)(π(g))−^1