Exotic Particles and $w_\infty$-Algebras in Two- and High-Dimensional Spaces

TL;DR

This paper constructs noncommuting translation operators in multi-dimensional lattices that form $w_{}$-algebras, preserving the braiding properties of exotic particles in two-dimensional space.

Contribution

It introduces a novel method to build $w_{}$-algebras using noncommuting link operators in higher-dimensional lattices, extending the understanding of exotic particle symmetries.

Findings

Established noncommuting translation operators form $w_{}$-algebras.

Preserved braiding properties of exotic particles in the construction.

Extended algebraic structures to high-dimensional lattice systems.

Abstract

We construct a set of noncommuting translation operators in two and high-dimensional lattices. The algebras they close are $w_{\infty}$-algebras. The construction is based on the introduction of noncommmuting elementary link operators which link two neighborhood sites in the lattice. This kind of operators preserve the braiding nature of exotic particles living basically in two-dimensional space.