QUANTUM DOUBLE AND DIFFERENTIAL CALCULI

We show that bicovariant bimodules as defined by Woronowicz are in one -to-one correspondence with the Drinfeld quantum double representation s. We then prove that a differential calculus associated to a bicovari ant bimodule of dimension n is connected to the existence of a particu lar (n + 1)-dimensional representation of the double. An example of bi covariant differential calculus on the nonquasitriangular quantum grou p E(q)(2) is developed. The construction is studied in terms of Hochsc hild cohomology and a correspondence between differential calculi and l-cocycles is proved. Some differences of calculi on quantum and finit e groups with respect to Lie groups are stressed.