Abstract
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
dim
V
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
C
P
dimV−1
. They have dimension
dimV
. For a classical Lie group we obtain an infinite family of non-standard calculi. General constructions for bicovariant calculi and their quantum tangent spaces are also obtained.