Geometry of Quantum Group Twists, Multidimensional Jackson Calculus and Regularization

Abstract

We show that R-matricies of all simple quantum groups have the properties which permit to present quantum group twists as transitions to other coordinate frames on quantum spaces. This implies physical equivalence of field theories invariant with respect to q-groups (considered as q-deformed space-time groups of transformations) connected with each other by the twists. Taking into account this freedom we study quantum spaces of the special type: with commuting coordinates but with q-deformed differential calculus and construct $GL_r(N)$ invariant multidimensional Jackson derivatives. We consider a particle and field theory on a two-dimensional q-space of this kind and come to the conclusion that only one (time-like) coordinate proved to be discretized.