Non-Trivial Extensions of the 3D-Poincaré Algebra and Fractional Supersymmetry for Anyons

Abstract

Non-trivial extensions of the three dimensional Poincaré algebra, beyond the supersymmetric one, are explicitly constructed. These algebraic structures are the natural three dimensional generalizations of fractional supersymmetry of order $F$ already considered in one and two dimensions. Representations of these algebras are exhibited, and unitarity is explicitly checked. It is then shown that these extensions generate symmetries which connect fractional spin states or anyons. Finally, a natural classification arises according to the decomposition of $F$ into its product of prime numbers leading to sub-systems with smaller symmetries.

17 pages, LaTex, the algebraic structure is given with more details, unitarity of the respresentation is explicitly checked, references has been added, one author has been added, to appear in Mod. Phys. Lett. A

References in corpus (5)

• Group Theoretical Foundations of Fractional Supersymmetry
• Geometrical foundations of fractional supersymmetry
• Local Fractional Supersymmetry for Alternative Statistics
• 2D Fractional Supersymmetry for Rational Conformal Field Theory. Application for Third-Integer Spin States
• Fractional Superspace Formulation of Generalized Mechanics

Cited by in corpus (15)

• Fractional Supersymmetry and Fth-Roots of Representations
• Intermediate-statistics quantum bracket, coherent state, oscillator, and representation of angular momentum (su(2)) algebra
• Finite-dimensional Lie algebras of order F
• Fractional supersymmetry and hierarchy of shape invariant potentials
• Field Theoretic Realizations for Cubic Supersymmetry
• Hopf algebras for ternary algebras
• Poincaré and sl(2) algebras of order 3
• Non-Abelian Fractional Supersymmetry in Two Dimensions
• Intensity Interferometry with Anyons
• Four dimensional cubic supersymmetry
• Higher spin swampland conjecture for massive AdS${}_3$ gravity
• Fractional Supersymmetry and F-fold Lie Superalgebras
• Fractional Supersymmetry and Infinite Dimensional Lie Algebras
• Some Results on Cubic and Higher Order Extensions of the Poincaré Algebra
• Scattering and Bound State Green's Functions on a Plane via so(2,1) Lie Algebra