Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1993-02-17
J.Math.Phys. 34 (1993) 4843-4856
Physics
High Energy Physics
High Energy Physics - Theory
19 pages
Scientific paper
10.1063/1.530326
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.
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