Let $\varGamma =\varGamma _{\tau ,z}$ be one of the $N^2$-dimensional bicovariant first order differential calculi for the quantum groups $\mathrm{GL}_q(N)$, $\mathrm{SL}_q(N)$, $\mathrm{SO}_q(N)$, or $\mathrm{Sp}_q(N)$, 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 $\varGamma^\land $ coincides with the de Rham cohomologies of its left-coinvariant, its right-coinvariant and its (two-sided) coinvariant subcomplexes. In the cases $\mathrm{GL}_q(N)$ and $\mathrm{SL}_q(N)$ the cohomology ring is isomorphic to the coinvariant external algebra $\varGamma ^\land _{\scriptscriptstyle{\mathrm{Inv}}}$ 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. 2000 Mathematical Subject Classification: 46L87, 58A12, 81R50.

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace–Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite-dimensional. Using Jucys–Murphy elements of the Hecke algebra, the eigenvalues of the Laplace–Beltrami operator for the Hopf algebra ${\mathcal {O}}$(SLq(N)) are computed.