Lars Tuset

Co-Amenability of Compact Quantum Groups

Abstract We study the concept of co-amenability for a compact quantum group. Several conditions are derived that are shown to be equivalent to it. Some consequences of co-amenability that we obtain are faithfulness of the Haar integral and automatic norm-boundedness of positive linear functionals on the quantum group’s Hopf ∗ -algebra (neither of these properties necessarily holds without co-amenability).

Amenability and Co-Amenability for Locally Compact Quantum Groups

We define concepts of amenability and co-amenability for locally compact quantum groups in the sense of J. Kustermans and S. Vaes. Co-amenability of a lcqg (locally compact quantum group) is proved to be equivalent to a series of statements, all of which imply amenability of the dual lcqg. Further, it is shown that if a lcqg is amenable, then its universal dual lcqg is nuclear. We also define and study amenability and weak containment concepts for representations and corepresentations of lcqgs.

Differential calculi over quantum groups and twisted cyclic cocycles

Abstract We study some aspects of the theory of non-commutative differential calculi over complex algebras, especially over the Hopf algebras associated to compact quantum groups in the sense of S.L. Woronowicz. Our principal emphasis is on the theory of twisted graded traces and their associated twisted cyclic cocycles. One of our principal results is a new method of constructing differential calculi, using twisted graded traces.

THE DIRAC OPERATOR ON COMPACT QUANTUM GROUPS

Abstract For the q-deformation Gq , 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element of U𝔤 ⊗ Cl(𝔤). The commutator of Dq with a regular function on Gq consists of two parts. One is a twist of a classical commutator and so is automatically bounded. The second is expressed in terms of the commutator of the associator with an extension of D. We show that in the case of the Drinfeld associator the latter commutator is also bounded.

On amenability and co-amenability of algebraic quantum groups and their corepresentations

We introduce and study several notions of amenability for unitary corepresentations and ∗-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background forthis study, we investigate the associated tensor C � -categories.

AMENABILITY AND COAMENABILITY OF ALGEBRAIC QUANTUM GROUPS

We define concepts of amenability and coamenability for algebraic quantum groups in the sense of Van Daele (1998). We show that coamenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or sufficient conditions for amenability or coamenability are obtained. Coamenability is shown to have interesting consequences for the modular theory in the case that the algebraic quantum group is of compact type.

Amenability and co-amenability of algebraic quantum groups II

Abstract We continue our study of the concepts of amenability and co-amenability for algebraic quantum groups in the sense of A. Van Daele and our investigation of their relationship with nuclearity and injectivity. One major tool for our analysis is that every non-degenerate ∗ -representation of the universal C ∗ -algebra associated to an algebraic quantum group has a unitary generator which may be described in a concrete way.

ON SECOND COHOMOLOGY OF DUALS OF COMPACT GROUPS

We show that for any compact connected group G the second cohomology group defined by unitary invariant 2-cocycles on \hat G is canonically isomorphic to H^2(\hat{Z(G)};T). This implies that the group of autoequivalences of the C*-tensor category Rep G is isomorphic to H^2(\hat{Z(G)};T)\rtimes\Out(G). We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof-Gelaki and Izumi-Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras. In two appendices we give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analogue of Radfords theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups.

Quantum groups, differential calculi and the eigenvalues of the Laplacian

We study �-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition of the calculus of forms, and, for Woronowicz’ first calculus, the calculation of the eigenvalues of the Laplacian. 1

Aspects of compact quantum group theory

We show that if a compact quantum semigroup satisfies certain weak cancellation laws, then it admits a Haar measure, and using this we show that it is a compact quantum group. Thus, we obtain a new characterization of a compact quantum group. We also give a necessary and sufficient algebraic condition for the Haar measure of a compact quantum group to be faithful, in the case that its coordinate C*-algebra is exact. A representation is given for the linear dual of the Hopf *-algebra of a compact quantum group, and a functional calculus for unbounded linear functionals is derived.