Resumen de L2-homology for compact quantum groups

David Kyed

• A notion of L2-homology for compact quantum groups is introduced, generalizing the classical notion for countable, discrete groups. If the compact quantum group in question has tracial Haar state, it is possible to define its L2-Betti numbers and Novikov-Shubin invariants/capacities. It is proved that these L2-Betti numbers vanish for the Gelfand dual of a compact Lie group and that the zeroth Novikov-Shubin invariant equals the dimension of the underlying Lie group. Finally, we relate our approach to the approach of A. Connes and D. Shlyakhtenko by proving that the L2-Betti numbers of a compact quantum group, with tracial Haar state, are equal to the Connes- Shlyakhtenko L2-Betti numbers of its Hopf .-algebra of matrix coefficients.