The Noncommutative Inhomogeneous Hopf Algebra

Abstract

From the bicovariant first order differential calculus on inhomogeneous Hopf algebra $\mathcal{B}$ we construct the set of right-invariant Maurer-Cartan one-forms considered as a right-invariant basis of a bicovariant $\mathcal{B}$-bimodule over which we develop the Woronowicz's general theory of differential calculus on quantum groups. In this formalism, we introduce suitable functionals on $\mathcal{B}$ 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 $\mathcal{B}$ and translations are controled by different R-matrices satisfying nontrivial characteristic equations.

25 pages, TeX, no figures. A section concerning the consistency conditions between the different functionals and the inhomogeneous commutation rules is added. Some references are also added