3
平移群：
考虑二维空间(x, y)。在此类空间中，平移由平移向量(Dx, Dy)定义。我们通常写作：
(27)
x' = x + Dx
y' = y + Dy
为了得到新的值x'和y'，我们使用加法。我们能否通过乘法获得相同的结果？
考虑以下矩阵：
(28)
注意到它们由两个独立参数Dx和Dy定义。因此，该群的维数为2。
形式：
(29)
注意到这与简单的矩阵乘法有本质区别：
(30)
g × r
这是一种特殊的群作用。
(31)
此外，我们也可以考虑三维或四维空间中的平移。相应的方阵构成的群为：
(32)
(33)
对应的作用为：
(34)
平移群是可交换的，其单位元是零平移。
矩阵群：为何需要？
利用矩阵群，我们可以将多个操作合并为单一操作、单一作用。考虑以下矩阵和作用：
(35-1)
我们结合了两件事：一个旋转（角度a），以及一个平移（Dx, Dy）。
群G中的元素g作用于空间r = (x, y)，并非“直接”作用，而是通过一种更精细的“作用”方式实现。
这个群
(35-2)
被称为“特殊欧几里得群SE(2)”，作用于二维空间。这一名称将在后文解释。
它的维数是多少？它依赖于三个自由参数：(a, Dx, Dy)，因此其维数为三。我们可以写作：
g ∈ SE(a, Dx, Dy)
子群。
对我们而言，群是一组方阵。在该集合中，我们可以找到一些子集。
g ∈ SE(0, Dx, Dy) 是平移子群。
g ∈ SE(a, 0, 0) 是绕原点0的旋转子群。
g ∈ SE(0, Dx, 0) 是平行于OX轴的平移子群。
上述群用于传递点。这些点没有特殊特征，它们仅仅是……点，仅此而已。
但稍后，描述物理世界的其他群将传递具有不同特征（“属性”）的点：质量、能量、动量、自旋等。
在上述群中，只有点的集合才值得被传递。此时，一个基本概念浮现出来：
种类。
我们的第一个群传递几何对象，即点的集合，几何图形（“刚性”）。最简单的集合由两个点组成。考虑二维空间中的点对：
(35-3)
在图(35-3)中，表示了两组点对(A, B)和(A', B')。我可以找到群中的一个元素，将(A, B)变换为(A', B')：通过绕点O旋转并结合平移。参见图(35-4)。
(35-4)
现在考虑两组点对：
(35-5)
无法找到群G中的任何元素g（方阵），能将(A, B)映射到(A", B")。我将说：
(A, B)和(A', B')属于同一种类。
(A, B)和(A", B")属于不同种类。
点对种类的特征称为“长度”。
这就是群论中对“长度”的定义。
你如何断言两个线段长度相等？因为你可以将它们进行比较，使一个叠在另一个上。
在我们的群中，长度不同的两条线段属于不同种类，因为我们的群不允许伸缩或压缩（相似变换）。负责处理这类变换的群是另一个群（“特殊笛卡尔群”）：
(35-6)
相对于这个群，所有点对都属于同一种类。该群的维数为四。
除了两个点，我们也可以考虑三个或四个点，例如构成正方形的情况。
(36)
相对于群(35-1)，边长相等的正方形属于同一种类。但如果两个正方形的边长本质不同：
(37)
它们就属于不同种类。
这个群控制二维平移以及平面内绕固定点的旋转，称为特殊欧几里得群：SE(2)。
我们可以很容易地想象一个类似的群作用于三维空间。三维和四维平移群已在(32)和(33)中给出。
我们也可以轻松设想一个描述n维空间中平移的群。但旋转又如何处理？
我们可以想象三维空间中的旋转，甚至可以用包含三个角度（欧拉角）的矩阵来表示：因此其维数为三。
索引 动态群论
原始英文版
3
Group of translations :
Consider 2d space (x,y). In such space a translation is defined by translation vector (
Dx,
Dy). We use to write :
(27)
x' = x +
D x
y' = y +
D y
To get the new values x' and y' we use
addition
. Could we get the same results through a .....
multiplication
?
Consider the following matrixes :
(28)
Notice they are defined by two independent parameters
Dx and
Dy. Then the dimension of the group is 2.
Form :
(29)
Notice this is basically different from the simple matricial multiplication
(30)
g x r
It is a peculiar group's action.
(31)
By the way, notice we can consider translations in 3d or 4d spaces. The corresponding square matrixes, forming groups, are
(32)
(33)
The corresponding action is :
(34)
The group of translations is commutative. Its neutral element is the null-translation.
Groups of matrixes : why ?
With matrixes' groups we can combine several operations into a single one, into a single action. Consider the following matrixes and the following action :
(35-1)
We combine two things : a rotation ( angle a ), plus a translation (
Dx,
Dy).
The element
g
of the group G acts on space
r
= (x,y), not "directly" but through some more refined "action".
This group
(35-2)
called "Special Euclid's group SE(2) ", acts on 2d space. This name will be explained further.
What is its dimension ? It depends on three free parameter : (
a
,
D x ,
Dy), so that its dimension is three. We may write :
g
SE (
a
,
Dx ,
D y)
Sub-groups.
For us, a group is a set of square matrixes. Among this set we can find sub-sets.
g
SE
(0,
Dx,
Dy) is the sub-group of translations.
g
SE
( a
, 0, 0) is the sub-group of rotations around the origin 0 .
g
SE
(0,
Dx, 0) is the sub-group of translation parallel to the axis OX.
The above group carries points. These point own no peculiar characteristics. They are... points, nothing else.
But, later, other groups, which describe physical world, will carry points which will have different characteristics, "attributes" : mass, energy, impulsion, spin....
With the above group only sets of points are interesting to carry. Here appears the fundamental concept of :
Species.
Our first group carries geometrical objects, which are sets of points, geometrical ("rigid") figures. The most simple set is composed by two points. Consider couples of points in a 2d space :
(35-3)
On figure (35-3) two couples of points (A,B) and (A',B') have been figured. I can find an element of the group that transforms (A,B) into (A',B') : combining a rotation around the point O and a translation. See figure (35-4).
(35-4)
Now consider the two couples :
(35-5)
Impossible to find any element
g
( square matrix ) of my group G which can carry (A,B) on (A",B"). I will say that:
(A,B) and (A',B') belong to a same species.
(A,B) and (A",B") belong to different species.
The characteristic of a species of couples of points is called
length
.
This is the definition of length in terms of group theory.
How can you affirm that two segments have the same length ? Because you can compare them, putting one onto the other one.
In our group two segments, whose lengths, are different belong to different species, because our group does not rule dilatations or contractions ( homothetic transforms ). The group which takes that in charge is a different one ("Special Descartes' group" ):
(35-6)
with respect to such group all couples of points form the same species. The dimension of this group is four.
Instead two points, we could consider three or four, these last forming squares, for an example.
(36)
With respect to the group (35-1), squares whose sides have the same length belong to the same species. But if the sides of two squares are basically different :
(37)
they belong to different species.
This group, ruling 2d translation and rotations around a fixed point of a plane is the Special Euclid's group : SE(2).
Now we imagine easily a similar group acting on a 3d space. The group of 3d and 4d translations were given in (32) , (33).
We can imagine easily a group describing translations in a n-dimensional space. But what about rotations ?
We can imagine rotation in a 3d space. We can even write it with a matrix which contains three angles, the Euler angles : then its dimension is three.
Index Dynamic Groups Theory