Poincar\'e superalgebras and triple systems

TL;DR

This paper classifies Poincaré superalgebras in various dimensions, characterizes their structure via embedding superalgebras, and introduces a construction method using triple systems for low-dimensional cases.

Contribution

It provides a complete classification of classical embedding superalgebras defining Poincaré superalgebras and introduces a novel construction using triple systems in low dimensions.

Findings

Classified all classical embedding superalgebras for Poincaré superalgebras.

Characterized Poincaré superalgebras in dimensions >3.

Constructed Poincaré superalgebras in dimensions 1-3 from triple systems.

Abstract

We consider a class of Poincar\'e superalgebras for which the nested bracket of three supercharges is necessarily zero only in dimensions greater than three. In lower dimensions, we give a precise characterisation of the data which encodes any such Poincar\'e superalgebra in terms of a more elementary embedding superalgebra. Up to isomorphism, we classify every classical embedding superalgebra that defines a Poincar\'e superalgebra. More generally, we show how to construct an embedding superalgebra in dimensions one, two and three from a certain type of triple system whose product structure encodes the nested bracket of three supercharges in the associated Poincar\'e superalgebra.