5
A square matrix of rank (n,n) acts on a column vector (n,0). We have seen that the 2D Euclidean group, referring to a space (x,y), does not involve actions on column vectors:
(51)
but rather on column vectors:
(52)
This represents an example of an action of the group on a space X with x ∈ X. There are infinitely many possible actions, even just the action of the group on itself. Actions are defined by axioms.
(53)
Considering the column vector:
(54)
where x represents, for example, the vectors:
(55)
(56)
satisfies the axioms of a group action. One can then perform a left multiplication of the square matrix representing a group element with a row matrix y, asking whether this also constitutes an action.
(57)
A_g(y) = y x g
The answer is no. This is not a group action: it does not satisfy the axioms given above. This is what I like to call an "anti-action," obeying the following "anti-axioms":
(58)
A mathematician would say there is no need to invoke these "anti-actions," and that there is only one set of axioms. Indeed. Similarly, what is considered an anti-action:
(59)
AA_g(m) = g⁻¹ x m x g
where m is a given vector, "an anti-action of the group element g on the matrix m," with g⁻¹ denoting the inverse matrix, can be treated as an action of the element g⁻¹.
Likewise, an anti-action is simply the dual of an action. I found it convenient to introduce this concept for didactic reasons.
Starting from a group of square matrices depending on n parameters p_i, one can construct matrices by differentiating all these parameters according to dp_i. The resulting matrices, filled with elements dp_i, do not form a group, but rather what is called the "tangent vector to the group":
dg
(its "Lie algebra," which, by the way, is not truly an algebra, but let's pass over that).
Thus, the group can act on the tangent vector dg near the group's identity element e through the "anti-action":
(60)
AA_g(m) = g⁻¹ x dg x g
(g = e)
We thus obtain the following scheme:
(61)
But an anti-action is the dual of an action. And whenever there is duality, there is conservation of a scalar product S.
Souriau therefore sought to construct a second group action—the action of the group on its space of moments. However, this action, known as the coadjoint or essential action, could not emerge directly. He had to go through this intermediary, which I call the "anti-action of the group on its tangent vector."
Thus, the desired action emerges as the dual of the anti-action of the group on its tangent vector. And the dual of an anti-action is an action, which can be written as:
(62)
A_g(J)
where J will be the "moment": a constellation of quantities that are attributes of a "material point," the action in question, called the coadjoint action, showing how these attributes change during motion.
There exists a group, which will be given later, that is an extension of the Galilean group, also given later, called the Bargmann group (1960). Applying this method to this group, one can construct its moment J_B and the way the group acts on it.
Souriau is accustomed to saying:
"The moment follows motion like its shadow."
A beautiful image, borrowed from his work "Grammaire de la Nature." Indeed, the material point moves through spacetime (x,y,z,t). As it does so, its attributes evolve, described by this coadjoint action of the group on its space of moments.
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