Lie subalgebras of the Weyl algebra. Lie algebras of order 3 and their application to cubic supersymmetry

TL;DR

This paper explores the structure of Lie subalgebras within the Weyl algebra and introduces a novel Lie algebra of order 3, leading to the development of a cubic supersymmetry model with unique properties and constraints.

Contribution

It presents the first detailed study of Lie subalgebras of the Weyl algebra and develops a new cubic supersymmetry framework extending the Poincaré algebra, including invariant Lagrangians and interaction analysis.

Findings

Classification of finite-dimensional Lie subalgebras of the Weyl algebra

Construction of a cubic supersymmetry (3SUSY) model extending Poincaré algebra

Interactions are forbidden by cubic supersymmetry invariance

Abstract

In the first part we present the Weyl algebra and our results concerning its finite-dimensional Lie subalgebras. The second part is devoted to a more exotic algebraic structure, the Lie algebra of order 3. We set the basis of a theory of deformations and contractions of these algebraic structures. We then concentrate on a particular such Lie algebra of order 3 which extends in a non-trivial way the Poincar\'e algebra, this extension being different of the supersymmetric extension. We then focus on the construction of a field theoretical model based on this algebra, the {\it cubic supersymmetry} ({\it 3SUSY}). For this purpose we obtain bosonic multiplets with whom we construct invariant Lagrangians. We then study the compatibility between this new symmetry and the abelian gauge symmetry. Furthermore, the analyse of possible interactions shows that interactions terms are not allowed by the cubic supersymmetry invariance. Finally we establish results regarding the extension in arbitrary dimensions of our model.