Quantum Mechanics for Mathematicians

More concretely, given a basisX 1 ,X 2 ,...,Xdof a Lie algebragof dimension
dwith structure constantscjkl, a representation is given by a choice ofdcomplex
ndimensional matricesφ(Xj) satisfying the commutation relations

[φ(Xj),φ(Xk)] =

∑d

l=1

cjklφ(Xl)

The representation is unitary when the matrices are skew-adjoint.
The notion of a Lie algebra representation is motivated by the fact that
the homomorphism property causes the mapπto be largely determined by its
behavior infinitesimally near the identity, and thus by the derivativeπ′. One
way to define the derivative of such a map is in terms of velocity vectors of
paths, and this sort of definition in this case associates to a representation
π:G→GL(n,C) a linear map

π′:g→M(n,C)

where

π′(X) =

d

dt

(π(etX))|t=0

GL(n,C)

1

π(etX)

t= 0

π′(X)

G

t= 0 1

etX

X

π′(X) =

d

dt

π(etX)|t=0

π

Figure 5.1: Derivative of a representationπ:G→GL(n,C), illustrated in
terms of “velocity” vectors along paths.

For the case ofU(1) we classified in theorem 2.3 all irreducible representa-
tions (homomorphismsU(1)→GL(1,C) =C∗) by looking at the derivative