Free braided differential calculus, braided binomial theorem, and the braided exponential map
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.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- October 1993
- DOI:
- 10.1063/1.530326
- arXiv:
- arXiv:hep-th/9302076
- Bibcode:
- 1993JMP....34.4843M
- Keywords:
-
- High Energy Physics - Theory;
- Mathematics - Quantum Algebra
- E-Print:
- 19 pages