Yang-Baxter matrices and differential calculi on quantum hyperplanes

Abstract

It is shown that any invertible matrix R that solves the Yang-Baxter equations generates a set of quantum hyperplanes where differential calculi can be defined. The number of such quantum hyperplanes is given by the number of different eigenvalues of the matrix R. Several examples of two-dimensional quantum hyperplanes and differential calculi are presented. The relations of quantum hyperplanes and differential calculi are covariant WRT the quantum groups defined by the matrix R. In the generic cases the exterior differential satisfies the condition d2=0.