Some Results on Cubic and Higher Order Extensions of the Poincar\'e Algebra

TL;DR

This paper explores higher order extensions of the Poincaré algebra, introducing Lie algebras of order F, and studies their implications in (1+2)-dimensional and D-dimensional space-times, including connections to anyons and invariant Lagrangians.

Contribution

It introduces Lie algebras of order F to extend the Poincaré algebra and analyzes their structures and physical implications in various space-time dimensions.

Findings

Higher order Poincaré extensions connect to relativistic anyons.

Infinite dimensional extensions induce new symmetries.

Invariant Lagrangians are constructed for cubic extensions.

Abstract

In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincar\'e algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the Poincar\'e algebra or Supersymmetry. Some general features on the so-called Wess-Zumino model (the simplest field theory invariant under Supersymmetry) are then given. We further introduce an additional algebraic structure called Lie algebras of order F, which naturally comprise the concepts of ordinary Lie algebras and superalgebras. This structure enables us to define various non-trivial extensions of the Poincar\'e algebra. These extensions are studied more precisely in two different contexts. The first algebra we are considering is shown to be an (infinite dimensional) higher order extension of the Poincar\'e algebra in $(1+2)-$dimensions and turns out to induce a symmetry which connects relativistic anyons. The second extension we are studying is related to a specific finite dimensional Lie algebra of order three, which is a cubic extension of the Poincar\'e algebra in $D-$space-time dimensions. Invariant Lagrangians are constructed.