@article {Heckenberger:December 2001:0011-4626:1342,
	author = "Heckenberger I.",
	author = "Schuler A.",
	title = "Laplace Operator and Hodge Decomposition for Quantum Groups and Quantum Spaces",
	journal = "Czechoslovak Journal of Physics",
	volume = "51",
	year = "December 2001",
	abstract = "<P> Let {\varGamma}={\varGamma}_{\pm,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's external algebra {\varGamma}^{\land } coincides with the de Rham cohomologies of its left-invariant, its right-invariant and its biinvariant subcomplexes. In the cases {\mathrm{GL}_q(N)} and {\mathrm{SL}_q(N)} the cohomology ring is isomorphic to the biinvariant external algebra {\varGamma}^\land_{\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. 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.</P>",
	pages = "1342-1347(6)",
	url = "http://www.ingentaconnect.com/content/klu/cjop/2001/00000051/00000012/00368438"
}