Group of translations by a fixed distance

[Mentz114]

I think I need this group ( if it exists) to help solve a physics problem.
Not being a mathematician, what I'd really like is a matrix representation
and a rule for getting the covariant derivative in the event of a broken symmetry.

I'd be much obliged if anyone can give me any information.

M.

 

[HallsofIvy]

To represent translation by a matrix multiplication, you will have to use a "projective geometry" representation: each point (x,y,z) is represented by (x,y,z,1) with the understanding that (ax, ay, az, a) (a not 0) represents the same point. Then a translation by <u, v, w> can be written as
$$\left(\begin{array}{ccccccc}1 && 0 && 0 && u \\ 0 && 1 && 0 && v\\ 0 && 0 && 1 && w \\ 0 && 0 && 0&& 1\end{array}\right)\left(\begin{array}{c}x \\ y \\ z \\ 1\end{array}\right)= \left(\begin{array}{c}x+ u \\ y+ v \\ z+ w \\ 1\end{array}\right)$$

 

[Mentz114]

HallsofIvy, thank you very much. This looks promising. Seems to form a group under multiplication, with an identity if u=v=w=0.

$$\left(\begin{array}{ccccccc}1 && 0 && 0 && u \\ 0 && 1 && 0 && v\\ 0 && 0 && 1 && w \\ 0 && 0 && 0&& 1\end{array}\right)\left(\begin{array}{ccccccc}1 && 0 && 0 && u \\ 0 && 1 && 0 && v\\ 0 && 0 && 1 && w \\ 0 && 0 && 0&& 1\end{array}\right) = \left(\begin{array}{ccccccc}1 && 0 && 0 && 2u \\ 0 && 1 && 0 && 2v\\ 0 && 0 && 1 && 2w \\ 0 && 0 && 0&& 1\end{array}\right)<br />$$