Kinematical superspaces

TL;DR

This paper classifies 43 types of N=1 d=4 kinematical and aristotelian Lie superalgebras with spatial isotropy, and describes their associated homogeneous superspaces, providing a comprehensive structural understanding.

Contribution

It introduces a quaternionic formalism to classify and analyze these superalgebras and their superspaces, including central extensions and geometric limits, expanding the understanding of supersymmetric structures.

Findings

43 isomorphism classes of Lie superalgebras identified

27 homogeneous superspaces classified, some with parameters

Invariants of low rank determined and geometric relations explored

Abstract

We classify $N{=}1$ $d=4$ kinematical and aristotelian Lie superalgebras with spatial isotropy, but not necessarily parity nor time-reversal invariance. Employing a quaternionic formalism which makes rotational covariance manifest and simplifies many of the calculations, we find a list of $43$ isomorphism classes of Lie superalgebras, some with parameters, whose (nontrivial) central extensions are also determined. We then classify their corresponding simply-connected homogeneous $(4|4)$-dimensional superspaces, resulting in a list of $27$ homogeneous superspaces, some with parameters, all of which are reductive. We determine the invariants of low rank and explore how these superspaces are related via geometric limits.