L^2-homology for compact quantum groups

Equivariant geometric K-homology for compact Lie group actions

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphism.

Restriction to compact subgroups in the cyclic homology of reductive p-adic groups

Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.

The homology of the groupoid of the self-similar dihedral group

We give an example of a locally compact effective Hausdorff, minimal ample groupoid such that its rational homology differs from the $K$-theory of its reduced groupoid $C^*$-algebra. Moreover, we prove that such example satisfies Matui's AH-conjecture.

On the zeroth L^2-homology of a quantum group

We prove that the zeroth L^2-Betti number of a compact quantum group vanishes unless the underlying C*-algebra is finite dimensional and that the zeroth L^2-homology itself is non-trivial exactly when the quantum group is coamenable.

紧量子群上Dirac算子的K-同调类

K-homology class of the Dirac operator on a compact quantum group

本研究基于Nagy的KK等价结论，证明紧李群q形变G_q上Dirac算子的K-同调类对应经典情形，并得到Drinfeld扭关于q连续的相关结果。

K-homology of certain group C*-algebras

Motivated by the search for new examples of ``noncommutative manifolds'', we study the noncommutative geometry (in the sense of Connes) of the group C*-algebras of various discrete groups. The examples we consider are the infinite dihedral group Z \times_σ $Z_2$ and the semidirect product group $Z \times_σ Z$. We present a unified treatment of the K-homology and cyclic cohomology of these algebras.