Knowlpedia

Definition

E(1)

E(1) as a separately identified real Lie group, with its defining operations and dimension.

E(1)\mathrm{E}(1) is the group of Euclidean isometries of R1\mathbb R^{1}. Each element is x↦Ax+bx\mapsto Ax+b with A∈O(1,R)A\in\mathrm{O}(1,\mathbb R) and b∈R1b\in\mathbb R^{1}, and

(A,b)(A′,b′)=(AA′,Ab′+b).(A,b)(A^{\prime},b^{\prime})=(AA^{\prime},Ab^{\prime}+b).
Dimensions and structure

This group has real dimension 11. This gives the of translations by the indicated . In dimension one the orthogonal part is {±1}, giving translations and reflections of the line.

Direct verification

The displayed product is composition of affine maps. Orthogonal matrices preserve the Euclidean distance, and any Euclidean isometry is a translation followed by an orthogonal map. Adding the translation dimension to n(n−1)/2 gives n(n+1)/2.