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  "body": "\n## Example\n\nLet $O(1, 3)$ be the [[Lorentz group|QWK3BYVJ9W]], acting on Minkowski space $\\R^{1,3} = \\R^4$.\nThe *Poincaré group* is the set of pairs $(\\Lambda, a)$ with $\\Lambda \\in O(1, 3)$ and $a \\in \\R^4$, acting on $\\R^4$ by\n$$(\\Lambda, a) \\cdot x = \\Lambda x + a,$$\na Lorentz transformation followed by a [[translation|087VJM5Q77]].\nComposing two such maps gives the group law\n$$(\\Lambda, a)(\\Lambda', a') = (\\Lambda \\Lambda', \\ \\Lambda a' + a).$$\nCheck: $(\\Lambda, a) \\cdot ((\\Lambda', a') \\cdot x) = \\Lambda(\\Lambda' x + a') + a = \\Lambda\\Lambda' x + (\\Lambda a' + a)$.\nThe identity is $(I, 0)$ and the inverse of $(\\Lambda, a)$ is $(\\Lambda^{-1}, -\\Lambda^{-1} a)$.\n\nSo the Poincaré group is the [[semidirect product|0H3WZXCZNF]] $\\R^4 \\rtimes O(1, 3)$, a [[Lie group|7JBRGCVJRB]] of dimension $4 + 6 = 10$.\n\n## Remarks\n\nThe Poincaré group is the group of all isometries of Minkowski space, that is, the bijections $f$ of $\\R^4$ with $Q(f(x) - f(y)) = Q(x - y)$ for the Minkowski form $Q$.\nThe translations form an abelian normal subgroup, so the Poincaré group is not semisimple.\nOften the name refers to the identity component $\\R^4 \\rtimes SO^+(1, 3)$.\nIts irreducible unitary representations, studied by Wigner, are labelled by mass and spin in physics.\n\n## Explanations\n\n### L1 human\n\nThe Poincaré group is the full symmetry group of special relativity.\nBesides rotations and boosts, it contains the shifts of space and time.\nIts $10$ dimensions count $4$ translations, $3$ rotations and $3$ boosts.\nThe laws of physics in flat space-time have the same form in every frame related by an element of this group.\n",
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