Laplace operator and Hodge decomposition for quantum groups and quantum spaces *)

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Let Γ = Γ±, be one of the N2-dimensional bicovariant first order differential calculi for the quantum groups GLq (N), SLq (N), SOq (N), or Spq (N), where q is a transcendental complex number and z is a regular parameter. It is shown that the de Rham cohomology of Woronowicz's external algebra coincides with the de Rham cohomologies of its left-invariant, its right-invariant and its biinvariant subcomplexes. In the cases GLq (N and SLq (N) the cohomology ring is isomorphic to the biinvariant external algebra 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. It is also applicable for quantum Euclidean spheres. The eigenvalues of the Laplace-Beltrami operator in cases of the general linear quantum group, the orthogonal quantum group, and the quantum Euclidean spheres are given.

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