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Poincaré group

In physics, the Poincaré group is the group of isometries of Minkowski space. 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.

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 Erlanger program, the geometry of Minkowski space is defined by the Poincaré group.

In component form, the Lie algebra of the Poincaré group satisfies

where P is a vector and M is a 2-form. See sign convention.

See Wigner's classification.