$q$-graded Heisenberg algebras and deformed supersymmetries

TL;DR

This paper introduces a q-grading framework for deformed Heisenberg algebras, extending supersymmetry concepts, and explores the special case when q approaches -1, revealing algebraic transformations between fermionic and bosonic modes.

Contribution

It develops a q-grading on deformed Heisenberg algebras and extends supersymmetry, including a novel analysis of the q→-1 limit linking fermions and bosons.

Findings

q-grading generalizes supersymmetry in deformed algebras

The q→-1 limit maps fermionic modes to bosonic modes

Algebraic structures are significantly altered in the q→-1 case

Abstract

The notion of $q$-grading on the enveloping algebra generated by products of q-deformed Heisenberg algebras is introduced for $q$ complex number in the unit disc. Within this formulation, we consider the extension of the notion of supersymmetry in the enveloping algebra. We recover the ordinary $\mathbb{Z}_2$ grading or Grassmann parity for associative superalgebra, and a modified version of the usual supersymmetry. As a specific problem, we focus on the interesting limit $q\to -1$ for which the Arik and Coon deformation of the Heisenberg algebra allows to map fermionic modes to bosonic ones in a modified sense. Different algebraic consequences are discussed.