Super-Poincaré algebra

In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symmetries), and are Lie superalgebras. Thus a super-Poincaré algebra is a Z₂-graded vector space with a graded Lie bracket such that the even part is a Lie algebra containing the Poincaré algebra, and the odd part is built from spinors on which there is an anticommutation relation with values in the even part.

The Poincaré algebra describes the isometries of Minkowski spacetime. From the representation theory of the Lorentz group, it is known that the Lorentz group admits two inequivalent complex spinor representations, dubbed ${\displaystyle 2}$ and ${\displaystyle {\overline {2}}}$. Taking their tensor product, one obtains ${\displaystyle 2\otimes {\overline {2}}=3\oplus 1}$; such decompositions of tensor products of representations into direct sums is given by the Littlewood-Richardson rule.

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