Euclidean group

It is a subgroup of the affine group $Aff(n)$. They are linear isometries composed with translations.

We have $E(n)={\mathbb{R}}^n\times O(n)$. Indeed is a semidirect product

To see it, taking into account section semidirect product#Remarks, you can think of the map

that associates to any $g\in E(n)$ the map $g$ itself composed with the translation that sends $g(0)$ to 0.

The Lie algebra

The Lie algebra of the Euclidean group $E(n)$, denoted $\mathfrak{e}(n)$, consists of two components:

• ${\mathbb{R}}^n$, corresponding to the translation part,
• $\mathfrak{o}(n)$, the Lie algebra of the orthogonal group $O(n)$, corresponding to the rotation part.

Thus, we have the semidirect sum

where elements are of the form $(\mathbf{v},A)$, with $\mathbf{v}\in {\mathbb{R}}^n$ (translations) and $A\in \mathfrak{o}(n)$ (infinitesimal rotations). The algebra is structured as a semidirect product, where rotations act on translations.

For the case $n=3$, the Lie algebra $\mathfrak{e}(3)$ can be described explicitly using the following commutation relations:

1. Rotations: The generators $J_a$ of the rotation group $\text{SO}(3)$ satisfy the commutation relations of $\mathfrak{so}(3)$:

   $$[J_a,J_b]=\sum _{c=1}^3{ϵ}_{abc}{J}_c,$$

   where ${ϵ}_{abc}$ is the Levi-Civita symbol, encoding the structure constants of the Lie algebra $\mathfrak{so}(3)$.

2. Translations: The translation generators $P_a$ commute with each other:

   $$[P_a,P_b]=0.$$

3. Action of Rotations on Translations: The rotation generators $J_a$ act on the translation generators $P_b$ in a way that reflects how translations transform as vectors under rotations:

   $$[J_a,P_b]=\sum _{c=1}^3{ϵ}_{abc}{P}_c.$$