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

Abstract

Let $V$ be a complex vector space with a non-degenerate symmetric bilinear form and $\mathbb S$ an irreducible module over the Clifford algebra $Cl(V)$ determined by this form. A supertranslation algebra is a $\mathbb Z$-graded Lie superalgebra $\mathfrak m=\mathfrak{m}_{-2}\oplus\mathfrak{m}_{-1}$, where $\mathfrak{m}_{-2}=V$ and $\mathfrak{m}_{-1}=\mathbb S\oplus\cdots\oplus\mathbb{S}$ is the direct sum of an arbitrary number $N\geq 1$ of copies of $\mathbb S$, whose bracket $[\cdot,\cdot]|_{\mathfrak{m}_{-1}\otimes \mathfrak{m}_{-1}}:\mathfrak{m}_{-1}\otimes\mathfrak{m}_{-1}\rightarrow\mathfrak{m}_{-2}$ is symmetric, $\mathfrak{so}(V)$-equivariant and non-degenerate (that is the condition "$s\in\mathfrak{m}_{-1}, [s,\mathfrak{m}_{-1}]=0$" 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 $\dim V\geq3$ and classify them in terms of super-Poincar\'e algebras and appropriate $\mathbb Z$-gradings of simple Lie superalgebras.