Poincaré algebra

Definition

Let $(V,\eta)$ be a finite-dimensional real Minkowski vector space, so $\eta$ has Lorentzian signature. Its Poincaré algebra is the semidirect-product Lie algebra

where the translation algebra $V$ is abelian and $\mathfrak{so}(V,\eta)$ acts through its defining representation. Its bracket is

For four-dimensional Minkowski space, this is $\mathfrak{iso}(1,3)$, with dimension $6+4=10$. It is the Lie algebra of the Poincaré group and of its identity component. Thus passing to the proper, orthochronous component does not change the Lie algebra.

For a nondegenerate form of arbitrary signature, the same semidirect-product formula defines the inhomogeneous orthogonal algebra $\mathfrak{iso}(p,q)$. The name “Poincaré algebra” is reserved here for Lorentzian signature.