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In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. It is a 10-dimensional noncompact Lie group. The abelian group of translations is a normal subgroup while the Lorentz group is a subgroup, the stabilizer of a point. That is, the full Poincaré group is the semidirect product of the translations and the Lorentz transformations.
Another way of putting it is the Poincaré group is a group extension of the Lorentz group by a vector representation of it.
Its positive energy unitary irreducible representations are indexed by mass (nonnegative number) and spin (integer or half integer), and are associated with particles in quantum mechanics.
In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as an homogeneous space for the group.
In component form, the Lie algebra of the Poincaré group satisfies
- <math>[P_\mu, P_\nu] = 0<math>
- <math>[M_{\mu\nu}, P_\rho] = \eta_{\mu\rho} P_\nu - \eta_{\nu\rho} P_\mu<math>
- <math>[M_{\mu\nu}, M_{\rho\sigma}] = \eta_{\mu\rho} M_{\nu\sigma} - \eta_{\mu\sigma} M_{\nu\rho} - \eta_{\nu\rho} M_{\mu\nu} + \eta_{\nu\sigma} M_{\mu\rho}<math>
where <math>P<math> is the generator of translation and <math>M<math> is the generator of Lorentz transformations. See sign convention.