groups and physics coadjoint action momentum
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3 - Third group axiom: Every element must have an inverse, denoted g⁻¹, defined by:
g × g⁻¹ = g⁻¹ × g = 1
In our example, this reads:
that is, b = -a, or:
g⁻¹(a) = g(-a)
...Here, computing the matrix inverse was obvious. But this is not always the case, by far. What, then, is required for every matrix in the considered set to have an inverse, i.e., to be invertible? It is necessary and sufficient that its determinant be non-zero (we refer the reader to their linear algebra course). A theorem states that the determinant of a product of matrices equals the product of their determinants. By definition, the determinant of a diagonal matrix equals the product of its diagonal entries. For example:
Consequences: the determinant of all identity matrices 1 equals 1. Therefore:
det(g) multiplied by det(g⁻¹) equals unity ¹ 0
consequence: a matrix with zero determinant cannot have an inverse, which would contradict its definition. Furthermore:
4 - Fourth group axiom: The composition operation must be associative:
( g₁ × g₂ ) × g₃ = g₁ × ( g₂ × g₃ )
This is always true...
Dimension of a group:
...A brief digression on the dimension of a group (of matrices), which has nothing to do with the rank of the matrices composing it, nor with the number of quantities forming the "space on which the group acts" (e.g., the 2D space (x,y) or the 4D spacetime (x,y)).
...Here we have an example of a family of square matrices depending on a single parameter a, which happens to form a group. Later, we will encounter groups made of square matrices defined by n parameters: six, ten, sixteen, or any number.
The number of parameters used to define the square matrices of the group will be called the dimension of the group.
We are dealing here with a group formed by a family of matrices depending on a single parameter a. The dimension of this group is 1.
Note in passing that:
Remark:
...Groups, and especially the groups we are interested in here, are not automatically commutative. In fact, commutativity is the exception. It happens that our example group is commutative:
...You will recognize in this group the 2D rotation matrices around a fixed axis. In "concrete" terms, this operation is "obviously commutative": rotating first by angle a, then by angle b, around an axis, yields the same result as rotating first by angle b, then by angle a.
You might say: "Of course. Rotation groups are essentially commutative."
...False. This is a property of 2D. It no longer holds in 3D. Consider a particular group formed by the set of rotations around three orthogonal axes (OX, OY, OZ).
Exercise: You will show, by taking an object and applying:
- First a +90° rotation around OX
- Then a +90° rotation around OZ
and then performing the same rotations in reverse order, that you do not obtain the same result. This operation is non-commutative.
Group action.
...A group G consists of a set of square matrices. We can already consider that it acts on itself (see below the axioms defining a group action, a fundamental concept).
...Our example group can also act on points in a "2D space". We say it rotates them. A group is meant to transport, but what exactly is being transported?
...Well, precisely, this is not what matters. Quoting his work "Grammar of Nature," we say with J.M. Souriau that:
The way of transporting is more important than what is being transported.
In the case of our example group, the matrices act on a 2D space (x,y), and we can write the corresponding action:
If we set (column matrix):
then the action is simply written as:
g × r
...In this particular case, the action of our group on the space (x,y) coincides with matrix multiplication. But we want to emphasize that this is merely a special case, and that the concept of action, fundamental in physics, is much more general.