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

Braided differential operators ∂ 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 expR(x|v) of the ∂ i (braided-plane waves) are introduced in the free case where the position components xi are totally noncommuting. We prove a braided R-binomial theorem and a braided-Taylors theorem expR(a|∂)f(x) = f(a + x). These various results precisely generalise to a generic R-matrix (and hence to n-dimensions) the well-known properties of the usual 1dimensional 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]>⊳C[p] of the braided line acting on itself (a braided Weyl algebra). Similarly for its generalization to an arbitrary R-matrix.