  Let G=G(t,z) be one of the N^2-dimensional bicovariant first order
differential calculi for the quantum groups GL_q(N), SL_q(N), O_q(N), or
Sp_q(N), where q is a transcendental complex number and z is a regular
parameter. It is shown that the de Rham cohomology of Woronowicz' external
algebra G^ coincides with the de Rham cohomologies of its left-coinvariant, its
right-coinvariant and its (twosided) coinvariant subcomplexes. In the cases
GL_q(N) and SL_q(N) the cohomology ring is isomorphic to the coinvariant
external algebra G^_{inv} and to the vector space of harmonic forms. We prove a
Hodge decomposition theorem in these cases. The main technical tool is the
spectral decomposition of the quantum Laplace-Beltrami operator.
  Keywords: quantum groups, bicovariant differential calculi, de Rham
cohomology, Laplace-Beltrami operator, Hodge theory
