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

Adrian Tanasă|ArXiv.org|Sep 22, 2005

Nonlinear Waves and Solitons3 citations

TL;DR

This paper classifies finite-dimensional Lie subalgebras of the Weyl algebra 𝒜₁ and introduces Lie algebras of order 3 as a generalization of Lie algebras, showing their application to cubic supersymmetry. It establishes a classification theorem for such subalgebras and constructs representations and deformations of elementary Lie algebras of order 3, linking them to cubic supersymmetry through the Weyl algebra framework.

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 Poincaré 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.

Motivation & Objective

• To classify finite-dimensional Lie subalgebras of the Weyl algebra 𝒜₁.
• To define and study Lie algebras of order 3 as a generalization of Lie algebras.
• To construct representations and deformations of elementary Lie algebras of order 3.
• To establish a connection between Lie algebras of order 3 and cubic supersymmetry.
• To explore the orbit structure of a family of Lie algebras under automorphism groups of 𝒜₁ and sl(2).

Proposed method

• Applies the Dixmier partition of the Weyl algebra to classify finite-dimensional Lie subalgebras.
• Uses the realization of sl(2) in 𝒜₁ as a foundational structure for constructing Lie algebras of order 3.
• Employs automorphism groups Aut(𝒜₁) and Aut(sl(2)) to analyze equivalence classes of Lie algebras.
• Introduces the concept of Lie algebras of order F, with F=3 as a special case.
• Applies deformation theory to study continuous families of Lie algebras of order 3.
• Utilizes matrix representations and polynomial realizations in 𝒜₁ to construct explicit models.

Experimental results

Research questions

• RQ1 Which finite-dimensional Lie algebras can be realized as subalgebras of the Weyl algebra 𝒜₁?
• RQ2 What is the complete classification of finite-dimensional Lie subalgebras of Der(𝒜₁)?
• RQ3 How do Lie algebras of order 3 generalize standard Lie algebras and what are their defining properties?
• RQ4 What is the orbit structure of a specific family of Lie algebras of order 3 under the action of Aut(𝒜₁) × Aut(sl(2))?
• RQ5 How can Lie algebras of order 3 be deformed and what are the implications for cubic supersymmetry?

Key findings

• A complete classification theorem is established for finite-dimensional Lie algebras realizable in the Weyl algebra 𝒜₁.
• The family F of Lie algebras of order 3 is fully characterized up to equivalence under Aut(𝒜₁) × Aut(sl(2)) and its isotropy group is determined.
• Representations of elementary Lie algebras of order 3 are constructed explicitly using polynomial realizations in 𝒜₁.
• Deformations of elementary Lie algebras of order 3 are studied, showing non-trivial continuous families.
• The paper establishes a direct link between Lie algebras of order 3 and cubic supersymmetry, providing a new algebraic framework for such symmetries.
• The orbit of the family F under Aut(𝒜₁) × Aut(sl(2)) is shown to be a complete invariant for classifying such Lie algebras.