What does Bourbaki mean by displacement in Lie Groups and Lie Algebras chapter 4-6

Question

In Bourbaki Lie Groups and Lie Algebras chapter 4-6 the term displacement is used a lot. For example groups generated by displacements. But I can not find a definition of the term displacement given anywhere. I also looked at Humphreys Reflection Groups and Coxeter groups book but I could not find it. Can someone provide a definiton of displacements in the context of reflection groups and root systems?

Answer 1

A displacement (déplacement in french) is a, affine isometry whose associated endomorphism (which is an isomorphism) is a rotation. In dimension $n$ it is isomorphic to the semi-direct product $\mathbf{R}^{n}\rtimes SO(\mathbf{R},n)$ as such an isometry is a composition of a rotation with a translation.

In Bourbaki, Groupes et algèbres de Lie, chapitre V, paragraphe 2, section 4, the term déplacement is indeed used, and as you should have seen while reading it, it refers to Bourbaki, Algèbre, chapitre IX, paragraphe 6, section 6, définition 3.

Answer 2

The group $E(n)=O_n(\Bbb R)\ltimes \Bbb R^n$ is the isometry group of the Euclidean space $\Bbb R^n$. It is also sometimes called the group of Euclidean motions of $\Bbb R^n$. According to wikipedia a displacement is an element of the isometry group $E^+(n)$. Note that $E^+(n)=SO_n(\Bbb R)\ltimes \Bbb R^n$ is the group of direct isometries, see wikipedia.