FREE BRAIDED DIFFERENTIAL-CALCULUS, BRAIDED BINOMIAL THEOREM, AND THE BRAIDED EXPONENTIAL MAP

Braided differential operators partial derivative(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 partial derivative(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\partial derivative)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]XC[p] of the braided line acting on itself (a braided Weyl algebra) and similarly for its generalization to an arbitrary R-matrix.