Classification of Bicovariant Differential Calculi

We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 1-1 correspondence with irreducible representations V of the quantum group enveloping algebra. The corresponding calculus is constructed and has dimension dimV 2. The differential calculi on a finite group algebra CG are also classified and shown to be in correspondence with pairs consisting of an irreducible representation V and a continuous parameter in CP dimV−1. They have dimension dimV. For a classical Lie group we obtain an infinite family of nonstandard calculi. General constructions for bicovariant calculi and their quantum tangent spaces are also obtained.