Free Braided Differential Calculus, Braided Binomial Theorem and the Braided Exponential Map
Abstract
Braided differential operators 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 R(|) of the i (braided-plane waves) are introduced in the free case where the position components xi are totally non-commuting. We prove a braided R-binomial theorem and a braided-Taylors theorem R(|)f()=f(+). 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 [x] [p] of the braided line acting on itself (a braided Weyl algebra). Similarly for its generalization to an arbitrary R-matrix.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.