Classification of maximal transitive prolongations of super-Poincaré 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 [·,·]|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.