TY - JOUR
AU - Kyed, David
PY - 2008/09/01
Y2 - 2024/07/14
TI - $L^2$-homology for compact quantum groups
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 103
IS - 1
SE - Articles
DO - 10.7146/math.scand.a-15072
UR - https://www.mscand.dk/article/view/15072
SP - 111-129
AB - A notion of $L^2$-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 $L^2$-Betti numbers and Novikov-Shubin invariants/capacities. It is proved that these $L^2$-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 $L^2$-Betti numbers of a compact quantum group, with tracial Haar state, are equal to the Connes-Shlyakhtenko $L^2$-Betti numbers of its Hopf $*$-algebra of matrix coefficients.
ER -