$q$-Deformed Classical Lie Algebras and their Anyonic Realization

TL;DR

This paper demonstrates how to realize $q$-deformed classical Lie algebras using anyonic oscillators, extending the fermionic oscillator approach and enabling applications on one-dimensional chains.

Contribution

It introduces a method to construct $q$-deformed Lie algebra representations with anyonic oscillators, generalizing fermionic realizations and applicable on 1D chains.

Findings

Realization of $q$-deformed Lie algebras with anyonic oscillators

Applicable to one-dimensional chains with real $q$

Provides a bosonization formula for generators

Abstract

All classical Lie algebras can be realized \`a la Schwinger in terms of fermionic oscillators. We show that the same can be done for their $q$-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two dimensional lattice. The deformation parameter $q$ is a phase related to the anyonic statistical parameter. A crucial r\^ole in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the underformed ones. The entire procedure works even on one dimensional chains; in such a case $q$ can also be real.