The inhomogeneous quantum groups from differential calculi with classical dimension

Abstract

From the bicovariant first-order differential calculus on inhomogeneous Hopf algebra ℬ we construct the set of right-invariant Maurer–Cartan one-forms considered as a right-invariant basis of a bicovariant ℬ-bimodule over which we develop the Woronowicz general theory of differential calculus on quantum groups. In this formalism, we introduce suitable functionals on ℬ which control the inhomogeneous commutation rules. In particular, we find that the homogeneous part of commutation rules between the translations and those between the generators of the homogeneous part of ℬ and translations are controlled by different R-matrices satisfying characteristic equations.