Quantum Mechanics for Mathematicians

5.2.1 Lie algebra of the orthogonal group

Recall that the orthogonal groupO(n) is the subgroup ofGL(n,R) of matrices
Ω satisfying ΩT= Ω−^1. We will restrict attention to the subgroupSO(n) of
matrices with determinant 1, which is the component of the group containing
the identity, with elements that can be written as

Ω =etX

These give a path connecting Ω to the identity (takingesX,s∈[0,t]). We
saw above that the condition ΩT= Ω−^1 corresponds to skew-symmetry of the
matrixX
XT=−X

So in the case ofG=SO(n), we see that the Lie algebraso(n) is the space of
skew-symmetric (XT=−X)nbynreal matrices, together with the bilinear,
antisymmetric product given by the commutator:

(X,Y)∈so(n)×so(n)→[X,Y]∈so(n)

The dimension of the space of such matrices will be

1 + 2 +···+ (n−1) =

n^2 −n

2

and a basis will be given by the matricesjk, withj,k= 1,...,n,j < kdefined
as

(jk)lm=







−1 ifj=l,k=m

+1 ifj=m,k=l

0 otherwise

(5.3)

In chapter 6 we will examine in detail then= 3 case, where the Lie algebra
so(3) isR^3 , realized as the space of antisymmetric real 3 by 3 matrices, with a
basis the three matrices 12 , 13 , 23.

5.2.2 Lie algebra of the unitary group

For the case of the groupU(n) the unitarity condition implies thatXis skew-
adjoint (also called skew-Hermitian), satisfying

X†=−X

So the Lie algebrau(n) is the space of skew-adjointnbyncomplex matrices,
together with the bilinear, antisymmetric product given by the commutator:

(X,Y)∈u(n)×u(n)→[X,Y]∈u(n)

Note that these matrices form a subspace ofCn

2
of half the dimension,
so of real dimensionn^2. u(n) is a real vector space of dimensionn^2 , but it