Quantum Mechanics for Mathematicians

for allg∈Gandh 1 ,h 2 ∈H, the groupGis said to act onHby automorphisms.
Each mapΦgis an automorphism ofH. Note that sinceΦgis an action ofG,
we haveΦg 1 g 2 = Φg 1 Φg 2.

When the groups are Lie groups, taking the derivativeφg:h→hof the map
Φg:H→Hat the identity ofHgives a Lie algebra automorphism, defined by

Definition(Lie algebra automorphisms).If an action of elementsgof a group
Gon a Lie algebrah
X∈h→φg(X)∈h

satisfies
[φg(X),φg(Y)] =φg([X,Y])

for allg∈GandX,Y∈h, the group is said to act onhby automorphisms.

Given an actionφgof a Lie groupGonh, we get an action of elementsZ∈g
onhby linear maps:

X→Z·X=

d

dt

(φetZ(X))|t=0 (16.16)

that we will often refer to as the infinitesimal version of the actionφgofGon
h. These maps satisfy

Z·[X,Y] =

d

dt

(φetZ([X,Y]))|t=0

=

d

dt

([φetZ(X),φetZ(Y)])|t=0

=[Z·X,Y] + [X,Z·Y]

and one can define

Definition(Lie algebra derivations).If an action of a Lie algebragon a Lie
algebrahby linear maps
X∈h→Z·X∈h

satisfies
[Z·X,Y] + [X,Z·Y] =Z·[X,Y] (16.17)

for allZ∈gandX,Y∈h, the Lie algebragis said to act onhby derivations.
The action of an elementZonhis a derivation ofh.

16.2.1 The adjoint representation and inner automorphisms

Any groupGacts on itself by conjugation, with

Φg(g′) =gg′g−^1

giving an action by automorphisms (these are called “inner automorphisms”).
The derivative at the identity of the map Φgis the linear map onggiven by the