Free Braided Differential Calculus, Braided Binomial Theorem and the Braided Exponential Map

TL;DR

This paper develops a braided differential calculus framework using Yang-Baxter matrices, introducing braided eigenfunctions, binomial and Taylor theorems, and connecting to braided Weyl algebras, generalizing quantum calculus concepts.

Contribution

It introduces braided differential operators, eigenfunctions, and theorems that extend quantum calculus to arbitrary R-matrices and higher dimensions.

Findings

Established a braided R-binomial theorem

Proved a braided Taylor expansion theorem

Connected the q-Heisenberg algebra to braided Weyl algebra

Abstract

Braided differential operators $\del^i$ are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang-Baxter matrix $R$. The quantum eigenfunctions $\exp_R(\vecx|\vecv)$ of the $\del^i$ (braided-plane waves) are introduced in the free case where the position components $x_i$ are totally non-commuting. We prove a braided $R$-binomial theorem and a braided-Taylors theorem $\exp_R(\veca|\del)f(\vecx)=f(\veca+\vecx)$. These various results precisely generalise to a generic $R$-matrix (and hence to $n$-dimensions) the well-known properties of the usual 1-dimensional $q$-differential and $q$-exponential. As a related application, we show that the q-Heisenberg algebra $px-qxp=1$ is a braided semidirect product $\C[x]\cocross \C[p]$ of the braided line acting on itself (a braided Weyl algebra). Similarly for its generalization to an arbitrary $R$-matrix.