Quantum Mechanics for Mathematicians

same Lie bracket operation on pairs of vectors. This operation is familiar in yet
another context, that of the cross-product of standard basis vectorsejinR^3 :

e 1 ×e 2 =e 3 ,e 2 ×e 3 =e 1 ,e 3 ×e 1 =e 2

We see that the Lie bracket operation

(X,Y)∈R^3 ×R^3 →[X,Y]∈R^3

that makesR^3 a Lie algebraso(3) is the cross-product on vectors inR^3.
So far we have three different isomorphic ways of putting a Lie bracket on
R^3 , making it into a Lie algebra:

1. IdentifyR^3 with antisymmetric real 3 by 3 matrices and take the matrix
   commutator as Lie bracket.


2. IdentifyR^3 with skew-adjoint, traceless, complex 2 by 2 matrices and take
   the matrix commutator as Lie bracket.


3. Use the vector cross-product onR^3 to get a Lie bracket, i.e., define

[v,w] =v×w

Something very special that happens for orthogonal groups only in dimension
n= 3 is that the vector representation (the defining representation ofSO(n)
matrices onRn) is isomorphic to the adjoint representation. Recall that any Lie
groupGhas a representation (Ad,g) on its Lie algebrag.so(n) can be identified
with the antisymmetricnbynmatrices, so is of (real) dimensionn

(^2) −n
2. Only for
n= 3 is this equal ton, the dimension of the representation on vectors inRn.
This corresponds to the geometrical fact that only in 3 dimensions is a plane (in
all dimensions rotations are built out of rotations in various planes) determined
uniquely by a vector (the vector perpendicular to the plane). Equivalently,
only in 3 dimensions is there a cross-productv×wwhich takes two vectors
determining a plane to a unique vector perpendicular to the plane.
The isomorphism between the vector representation (πvector,R^3 ) on column
vectors and the adjoint representation (Ad,so(3)) on antisymmetric matrices is
given by 

v 1
v 2
v 3



↔v 1 l 1 +v 2 l 2 +v 3 l 3 =





0 −v 3 v 2

v 3 0 −v 1

−v 2 v 1 0





or in terms of bases by
ej↔lj
For the vector representation on column vectors,πvector(g) =gandπ′vector(X) =
X, whereXis an antisymmetric 3 by 3 matrix, andg=eXis an orthogonal 3
by 3 matrix. Both act on column vectors by the usual multiplication.