Poincaré group

The isometry group of Minkowski space: translations semidirect the Lorentz group.

Poincaré group

Let $\Bbb R^{1,3}$ denote Minkowski space with its standard bilinear form of signature $(1,3)$ .

Definition

The Poincaré group is the group of affine isometries of Minkowski space. Concretely, it is the semidirect product

\mathrm{ISO}(1,3)=\Bbb R^{1,3}\rtimes O(1,3),

where $O(1,3)$ is the Lorentz group acting linearly on $\Bbb R^{1,3}$ .

An element is a pair $(a,\Lambda)$ with $a\in \Bbb R^{1,3}$ (a translation) and $\Lambda\in O(1,3)$ , with group law

(a,\Lambda)\,(b,\Gamma)=(a+\Lambda b,\ \Lambda\Gamma).

A standard matrix model realizes $\mathrm{ISO}(1,3)$ as block matrices of the form

\begin{pmatrix} \Lambda & a\\ 0 & 1 \end{pmatrix}, \quad \Lambda\in O(1,3),\ a\in \Bbb R^{1,3},

with multiplication matching the semidirect product rule.

Lie algebra

Its Lie algebra is a semidirect sum

\mathfrak{iso}(1,3)=\Bbb R^{1,3}\rtimes \mathfrak{so}(1,3),

where $\mathfrak{so}(1,3)$ is the Lorentz Lie algebra (an instance of orthogonal Lie algebras ). Writing elements as pairs $(v,X)$ with $v\in \Bbb R^{1,3}$ and $X\in \mathfrak{so}(1,3)$ , the bracket is

[(v,X),(w,Y)] = (Xw-Yv,\ [X,Y]).

Action and orbits

The Poincaré group acts transitively on Minkowski space by affine transformations, so Minkowski space is a homogeneous space for $\mathrm{ISO}(1,3)$ . Stabilizers and orbits are central tools in representation theory and in the geometry of relativistic symmetries.