Introduction to Braided Geometry and $q$-Minkowski Space

TL;DR

This paper introduces the geometry of linear braided spaces with braid-statistics, surveys the braided approach to $q$-deformation including differentiation and integration, and presents natural $q$-Euclidean and $q$-Minkowski spaces.

Contribution

It provides a systematic introduction to braided geometry and applies it to $q$-deformed spaces, including $q$-Euclidean and $q$-Minkowski spaces, with foundational tools for $q$-physics.

Findings

Develops the theory of linear braided spaces with braid-statistics.

Surveys the braided approach to $q$-deformation including differentiation and integration.

Introduces natural $q$-Euclidean and $q$-Minkowski spaces in R-matrix form.

Abstract

We present a systematic introduction to the geometry of linear braided spaces. These are versions of $\R^n$ in which the coordinates $x_i$ have braid-statistics described by an R-matrix. From this starting point we survey the author's braided-approach to $q$-deformation: braided differentiation, exponentials, Gaussians, integration and forms, i.e. the basic ingredients for $q$-deformed physics are covered. The braided approach includes natural $q$-Euclidean and $q$-Minkowski spaces in R-matrix form.