6
Components of a group.
We have considered two groups: SO(2) and O(2). The second contains the first.
The first contains the neutral element. We can represent the elements of the group as follows:
(73)
.
The elements of the first component form a group (a subgroup).
The elements of the second group do not form a group, for many reasons:
-
It does not contain
the neutral element 1. -
We can choose two matrices from this second component, whose product does not belong to this second component. Example:
(74)
The component of the group containing the neutral element
1
is called
the neutral component of the group.
In the following, we will consider groups with 2, 4, 8 components.
The Euclidean group.
We can now integrate this extended, enriched group, to 2D translation, and we obtain:
(75)
as well as the corresponding action of this Euclidean group:
(76)
Suppose we use our group to manipulate, to rule, to study the letters of the alphabet.
Limit the set to:
A B C D E F G J K L N P Q R S Z
We have several sizes:
(77)
A B C D E F G J K L N P Q R S Z
A B C D E F G J K L N P Q R S Z
A B C D E F G J K L N P Q R S Z
We know that we cannot find any element of the group, nor any subsequent group action, capable of transforming:
G into G
because their sizes are different. We decide to call these sizes
masses
so that G and G are analogous to particles, objects, atoms possessing different masses.
Now, it depends on the group acting on this set of objects. If I use:
(78)
suppose this "world" is filled with:
(79)
with a certain spectrum of sizes (masses) and angles. If I perform group actions, whatever they are, I will never find objects belonging to the Russian alphabet:
(80)
It will become possible if I use the enriched group, the Euclidean group:
(80b)
Then my "world" will become:
(81)
The group has enriched the "zoo" of letters. But in my zoo, one is invariant under symmetry, that is:
(82)
(83)
(84)
(85)
In general, any symmetry with respect to any straight line of the plane, that is a "2D mirror", does not change the "nature" of this character
(86)
I will call this character a "photon" and assimilate the transformation
(87)
to the matter-antimatter duality. Thus, I obtain a global zoo:
(88)
We could link letters of the same shape (nature) but different sizes (representing their energies), using Descartes' group:
(89)
But we will not build a complete analogical model of elementary particles, based on alphabetical characters. Anyway, you begin to see where we are heading. Groups have very simple aspects, but hidden properties. These properties depend on their subgroups, which generate
species.
The Euclidean group is accompanied by an Euclidean world, an Euclidean zoo. The animals of Euclidean geometry are called sphere, cylinder, prisms, plane, straight line, triangles, etc. They are invariant under the action of a certain subgroup. Souriau calls the subgroup linked to an object belonging to a species the
regularity
of this object.
For example, spheres centered at a given point O are invariant under the subgroup of rotations around this point.
-
We can consider that
the fact of being invariant is a property of the species called "spheres centered at a point O". -
Conversely, we can consider that
this property
defines
the species.
Index Dynamic Groups Theory