Grassmann variables on quantum spaces

TL;DR

This paper develops techniques for antisymmetrized quantum spaces, including formulae for supernumber multiplication, symmetry actions, and Hopf structures, advancing the mathematical framework for quantum space analysis.

Contribution

It introduces new methods for handling q-deformed Grassmann variables and provides explicit formulas for symmetry actions and Hopf algebra structures in quantum spaces.

Findings

Formulas for multiplying supernumbers in quantum spaces

Explicit actions of symmetry generators and derivatives

Complete Hopf algebra structures for various quantum spaces

Abstract

Attention is focused on antisymmetrized versions of quantum spaces that are of particular importance in physics, i.e. two-dimensional quantum plane, q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. For each case standard techniques for dealing with q-deformed Grassmann variables are developed. Formulae for multiplying supernumbers are given. The actions of symmetry generators and fermionic derivatives upon antisymmetrized quantum spaces are calculated. The complete Hopf structure for all types of quantum space generators is written down. From the formulae for the coproduct a realization of the L-matrices in terms of symmetry generators can be read off. The L-matrices together with the action of symmetry generators determine how quantum spaces of different type have to be fused together.