Poincaré group

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

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

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

with multiplication matching the semidirect product rule.

Lie algebra

Its Lie algebra is a semidirect sum

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

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.