a4115

We need :
(180)

Ag
( Ag'(y)) = Ag"(y)

Ag(y)
= y x
g

Ag'(y)
= y x
g'

Ag
( Ag'(y)) = y x
g' x g

But :

The product of two matrixes
is not commutative, in general. As a conclusion :
(181)

Ag(y)
= y x g

is not a group's action
: It does not fit the above axioms. But it fits another, corresponding to
an "anti-action".
(182)

For matrixes :

(183

We are still searching
actions and anti-actions. From the x vector we can build the transposed
one and try :
(184)

is it an action
? Let us go.

g" = g
x g'

(185)

(186)

Here we use a theorem of linear calculus :
(187)

M -1
x N
= ( N x
M ) -1

where M et N
are arbitrary (n,n) matrixes. Whence :
(188)

g' -1
x g-1 = ( g x
g' ) -1
= g" -1

and :
(189)

which is a group's
action. Consider
now :
(190)

Ag(m)
= g x mx g-1

Show it is an action.
We shall consider the three following matrixes.
(191)

g

g'

g" = g
x g'

Ag(m)
= g x
m x g-1

Ag'(m)
= **g'**x
m x
g' -1

Ag"(m)
= g" x
m x
g" -1

We need :
(192)

Ag(Ag'(m))
= Ag"(m)

Form the first member
:
(193)

g x (g'
x m
x g'
-1) x g-1

or :
(194)

g x
g'x
m x
g' -1
x
g -1

i.e :
(195)

(g x
g') x
m x ( g
x g')-1 = g" x
m x
g" -1

It is a group's action.
We will call it, after Souriau :
Adjoint action
:
(193)

We are now going to consider an anti-action of the group on a matrix m.

(194)

AAg(m)
= g -1 x
m x
g

Show it satisfies :

(195)

AAg'(Ag(m))
= Ag"(m)

Form the first member
:
(196)

g'-1 x
(g -1 x
m x
g) x
g'

or :
(197)

g' -1
x
g -1
x
m x
g x
g'

i.e :
(198)

(g x
g') -1
x
m x
( g x
g')

or :
(199)

g" -1
x
m x
g"

Index Dynamic Groups Theory