De Rham Cohomology and Hodge Decomposition For Quantum Groups |
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Authors: | Heckenberger Istvan; Schuler Axel |
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Institution: | Department of Mathematics, University of Leipzig Augustusplatz 10, 04109 Leipzig, Germany, heckenbe{at}mathematik.uni-leipzig.de, schueler{at}mathematik.uni-leipzig.de |
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Abstract: | Let be one of the N2-dimensionalbicovariant first order differential calculi for the quantumgroups GLq(N), SLq(N), SOq(N), or Spq(N), where q is a transcendentalcomplex number and z is a regular parameter. It is shown thatthe de Rham cohomology of Woronowicz' external algebra coincides with the de Rham cohomologiesof its left-coinvariant, its right-coinvariant and its (two-sided)coinvariant subcomplexes. In the cases GLq(N) and SLq(N) thecohomology ring is isomorphic to the coinvariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in thesecases. The main technical tool is the spectral decompositionof the quantum Laplace-Beltrami operator. 2000 MathematicalSubject Classification: 46L87, 58A12, 81R50. |
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Keywords: | quantum groups bicovariant differential calculi de Rham cohomology Laplace-Beltrami operator Hodge theory |
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