Abstract
LetΓ=Γ ±,z be one of theN 2-dimensional bicovariant first order differential calculi for the quantum groups GL q (N), SL q (N), SO q (N), or Sp q (N), whereq is a transcendental complex number andz is a regular parameter. It is shown that the de Rham cohomology of Woronowicz’s external algebraΓ ^ coincides with the de Rham cohomologies of its leftinvariant, its right-invariant and its biinvariant subcomplexes. In the cases GL q (N) and SL q (N) the cohomology ring is isomorphic to the biinvariant external algebraΓ ^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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Heckenberger, I., Schüler, A. Laplace operator and hodge decomposition for quantum groups and quantum spaces. Czech J Phys 51, 1342–1347 (2001). https://doi.org/10.1023/A:1013326204779
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DOI: https://doi.org/10.1023/A:1013326204779