De Rham cohomology and Hodge decomposition for quantum groups (Q2766437)

The de Rham cohomology and the Hodge decomposition of the bicovariant differential calculi on the quantum groups of types A, B, C and D are studied. The Woronowicz construction of the external algebra is used. The first main theorem states that the embeddings of the left-coinvariant, the right-coinvariant, and the coinvariant subcomplexes in the whole complex of differential forms are quasi-isomorphisms, and, consequently, their de Rham cohomologies coincide. A Hodge decomposition theorem for \(SL_q(N)\) and \(GL_q(N)\) is the second main result. The technical tool is the quantum Laplace-Beltrami operator constructed by means of the dual pairing of two bicovariant differential calculi.