De Rham Cohomology and Hodge Decomposition For Quantum Groups

Let Γ = Γ τ , zbe one of the N 2 ‐dimensional bicovariant first order differential calculi for the quantum groups GL q ( N ), SL q ( N ), SO q ( N ), or 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' external algebra Γ ∧ coincides with the de Rham cohomologies of its left‐coinvariant, its right‐coinvariant and its (two‐sided) coinvariant subcomplexes. In the cases GL q ( N ) and SL q ( N ) the cohomology ring is isomorphic to the coinvariant external algebra ΓI n v∧ 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. 2000 Mathematical Subject Classification : 46L87, 58A12, 81R50.