ALGEBRAIC Q-INTEGRATION AND FOURIER THEORY ON QUANTUM AND BRAIDED SPACES

An algebraic theory of integration on quantum planes and other braided spaces is introduced. In the one-dimensional case a novel picture of the Jackson q- integral as indefinite integration on the braided group of functions in one variable x is obtained. Here x is treated with br aid statistics q rather than the usual bosonic or Grassmann ones. It i s shown that the definite integral integral(-x infinity)(x infinity) c an also be evaluated algebraically as multiples of the integral of a q -Gaussian, with x remaining as a bosonic scaling variable associated w ith the q-deformation. Further composing the algebraic integration wit h a representation then leads to ordinary numbers for the integral. In tegration is also used to develop a full theory of q-Fourier transform ation F The braided addition Delta x=xx1+1xx and braided-antipode S is used to define a convolution product, and prove a convolution theorem . It is also proven that F-2=S. The analogous results are proven on an y braided group, including integration and Fourier transformation on q uantum planes associated to general R matrices, including q-Euclidean and q-Minkowski spaces. (C) 1994 American Institute of Physics.