Anyons and Quantum Groups

TL;DR

This paper constructs anyonic oscillators with fractional statistics on a 2D lattice, explores their deformed commutation relations, and uses them to generate the quantum group SU(2)_q, linking anyons to quantum group symmetries.

Contribution

It introduces a generalized Jordan-Wigner construction for anyonic oscillators and demonstrates their role in generating quantum groups with fractional statistics.

Findings

Constructed anyonic oscillators on a lattice with fractional statistics.

Analyzed the deformed commutation relations of these anyonic oscillators.

Used the anyonic oscillators to generate the quantum group SU(2)_q.

Abstract

Anyonic oscillators with fractional statistics are built on a two-dimensional square lattice by means of a generalized Jordan-Wigner construction, and their deformed commutation relations are thoroughly discussed. Such anyonic oscillators, which are non-local objects that must not be confused with $q$-oscillators, are then combined \`a la Schwinger to construct the generators of the quantum group $SU(2)_q$ with $q=\exp({\rm i}\pi\nu)$, where $\nu$ is the anyonic statistical parameter.