Classification of maximal transitive prolongations of super-Poincaré algebras

Abstract

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.