Classification of maximal transitive prolongations of super-Poincar{\'e} algebras

Let V be a complex vector space with a non-degenerate symmetric bilinear form and S an irreducible module over the Clifford algebra Cℓ(V) determined by this form. A supertranslation algebra is a Z-graded Lie superalgebra m=m-2⊕m-1, where m-2=V and m-1=S⊕⋯⊕S is the direct sum of an arbitrary number N ≥ 1 of copies of S, whose bracket [{dot operator},{dot operator}]|m-1⊗m-1:m-1⊗m-1→m-2 is symmetric, so(V)-equivariant and non-degenerate (that is the condition ". s∈m-1,[s,m-1]=0" implies s = 0). We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finite-dimensional for dim <> V ≥ 3 and classify them in terms of super-Poincaré algebras and appropriate Z-gradings of simple Lie superalgebras. © 2014 Elsevier Inc.