Non-Trivial Extensions of the 3D-Poincar\'e Algebra and Fractional Supersymmetry for Anyons

TL;DR

This paper constructs and analyzes new algebraic structures extending the 3D Poincaré algebra, generalizing fractional supersymmetry to connect anyons with fractional spin, and classifies these extensions based on prime factorization of their order.

Contribution

It introduces novel non-trivial algebraic extensions of the 3D Poincaré algebra that generalize fractional supersymmetry and explores their representations and symmetry properties.

Findings

Extensions connect fractional spin states and anyons.

Representations are explicitly constructed and shown to be unitary.

Classification based on prime factorization of the order $F$.

Abstract

Non-trivial extensions of the three dimensional Poincar\'e 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.