Laplace Operator and Hodge Decomposition for Quantum Groups and Quantum Spaces

Authors: Heckenberger I.¹; Schüler A.²

Source: Czechoslovak Journal of Physics, Volume 51, Number 12, December 2001 , pp. 1342-1347(6)

Publisher: Springer

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Abstract:

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.

Language: English

Document Type: Regular paper

Affiliations: 1: Institute of Mathematics, University of Leipzig, Augustusplatz 10, 04109 Leipzig, Germany 2: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany

Publication date: 2001-12-01