Free braided differential calculus, braided binomial theorem, and the braided exponential map

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 exp_R(x‖v) of the ∂ⁱ (braided‐plane waves) are introduced in the free case where the position components x_i are totally noncommuting. A braided R‐binomial theorem and a braided Taylor theorem exp_R(a‖∂)f(x)=f(a+x) are proven. These various results precisely generalize to a generic R‐matrix (and hence to n dimensions) the well‐known properties of the usual one‐dimensional q‐differential and q‐exponential. As a related application, it is shown 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) and similarly for its generalization to an arbitrary R‐ matrix.