Quantum Mechanics for Mathematicians

Definition(Lie bracket). The Lie bracket operation ongis the bilinear anti-
symmetric map given by the commutator of matrices

[·,·] : (X,Y)∈g×g→[X,Y] =XY−Y X∈g

We need to check that this is well-defined, i.e., that it takes values ing.

Theorem.IfX,Y∈g,[X,Y] =XY−Y X∈g.

Proof.SinceX∈g, we haveetX∈Gand we can act onY∈gby the adjoint
representation
Ad(etX)Y=etXY e−tX∈g

Astvaries this gives us a parametrized curve ing. Its velocity vector will also
be ing, so
d
dt

(etXY e−tX)∈g

One has (by the product rule, which can easily be shown to apply in this case)

d

dt

(etXY e−tX) =

(

d

dt

(etXY)

)

e−tX+etXY

(

d

dt

e−tX

)

=XetXY e−tX−etXY Xe−tX

Evaluating this att= 0 gives
XY−Y X

which is thus, from the definition, shown to be ing.

The relation
d
dt

(etXY e−tX)|t=0= [X,Y] (5.1)

used in this proof will be continually useful in relating Lie groups and Lie alge-
bras.
To do calculations with a Lie algebra, one can choose a basisX 1 ,X 2 ,...,Xn
for the vector spaceg, and use the fact that the Lie bracket can be written in
terms of this basis as

[Xj,Xk] =

∑n

l=1

cjklXl (5.2)

wherecjklis a set of constants known as the “structure constants” of the Lie
algebra. For example, in the case ofsu(2), the Lie algebra ofSU(2) has a basis
X 1 ,X 2 ,X 3 satisfying

[Xj,Xk] =

∑^3

l=1

jklXl

(see equation 3.5) so the structure constants ofsu(2) are the totally antisym-
metricjkl.