Simple modules over finite quantum groups and their Drinfel'd doubles

Abstract

By finite quantum groups we mean Lusztig's finite-dimensional pointed Hopf algebras called quantum Frobenius Kernels [11,12], and their natural generalizations due to Andruskiewitsch and Schneider [2,3]. For a Hopf algebra H in a special class of the latter generalizations, which arises from a pair of quantum linear spaces, Krop and Radford [10] described the simple modules over H and over the Drinfel'd double D(H), showing that they fall into a simple pattern of parametrization. We extend the description to a wider class of Hopf algebras which includes the quantum Frobenius Kernels, renewing the parametrization pattern so as to connect directly to the so-called triangular decomposition.