Quantum Mechanics for Mathematicians

3. Recall 5.1:
   [X,Y] =
   d
   dt

(etXY e−tX)|t=0

so

π′([X,Y]) =π′

(

d

dt

(etXY e−tX)|t=0

)

=

d

dt

π′(etXY e−tX)|t=0 (by linearity)

=

d

dt

(π(etX)π′(Y)π(e−tX))|t=0 (by 2.)

=

d

dt

(etπ

′(X)

π′(Y)e−tπ

′(X)

)|t=0 (by 1.)

= [π′(X),π′(Y)]

This theorem shows that we can study Lie group representations (π,V)
by studying the corresponding Lie algebra representation (π′,V). This will
generally be much easier since theπ′are linear maps. Unlike the non-linear
mapsπ, the mapπ′is determined by its value on basis elementsXjofg. The
π′(Xj) will satisfy the same bracket relations as theXj(see equation 5.2). We
will proceed in this manner in chapter 8 when we construct and classify all
SU(2) andSO(3) representations, finding that the corresponding Lie algebra
representations are much simpler to analyze. Note though that representations
of the Lie algebragdo not necessarily correspond to representations of the group
G(when they do they are called “integrable”). For a simple example, looking
at the proof of theorem 2.3, one gets unitary representations of the Lie algebra
ofU(1) for any value of the constantk, but these are only representations of
the groupU(1) whenkis integral.
For any Lie groupG, we have seen that there is a distinguished representa-
tion, the adjoint representation (Ad,g). The corresponding Lie algebra represen-
tation is also called the adjoint representation, but written as (Ad′,g) = (ad,g).
From the fact that
Ad(etX)(Y) =etXY e−tX

we can differentiate with respect totand use equation 5.1 to get the Lie algebra
representation

ad(X)(Y) =

d

dt

(etXY e−tX)|t=0= [X,Y] (5.4)

This leads to the definition:

Definition(Adjoint Lie algebra representation).(ad,g)is the Lie algebra rep-
resentation given by
X∈g→ad(X)

wheread(X)is defined as the linear map fromgto itself given by

Y→[X,Y]