Autoequivalences of the tensor category of Uq(g)-modules

We prove that for q\in\C* not a nontrivial root of unity the cohomology group defined by invariant 2-cocycles in a completion of Uq(g) is isomorphic to H^2(P/Q;\T), where P and Q are the weight and root lattices of g. This implies that the group of autoequivalences of the tensor category of Uq(g)-modules is the semidirect product of H^2(P/Q;\T) and the automorphism group of the based root datum of g. For q=1 we also obtain similar results for all compact connected separable groups.Comment: 5 pages; minor corrections; corollary about Drinfeld twists adde

Traces on crossed products

We give a description of traces on C(X)\rtimes G in terms of measurable fields of traces on the C*-algebras of the stabilizers of the action of G on X.Comment: 5 pages, AMS-LaTe

Classification of non-Kac compact quantum groups of SU(n) type

We classify up to isomorphism all non-Kac compact quantum groups with the same fusion rules and dimension function as $SU(n)$. For this we first prove, using categorical Poisson boundary, the following general result. Let $G$ be a coamenable compact quantum group and $K$ be its maximal quantum subgroup of Kac type. Then any dimension-preserving unitary fiber functor $Rep\ G \to Hilb_f$ factors, uniquely up to isomorphism, through $Rep\ K$. Equivalently, we have a canonical bijection $H^2(\hat G; T) \cong H^2(\hat K; T)$. Next, we classify autoequivalences of the representation categories of twisted $q$-deformations of compact simple Lie groups.Comment: 22 pages; v1: subsumes and strengthens the classification result from arXiv:1310.4407; v2: minor improvements, appendix corrected; v3: minor corrections, final versio

The variational principle for a class of asymptotically abelian C*-algebras

Let (A,\alpha) be a C*-dynamical system. We introduce the notion of pressure P_\alpha(H) of the automorphism \alpha at a self-adjoint operator H\in A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense \alpha-invariant *-subalgebra \A of A such that for all pairs a,b\in\A the C*-algebra they generate is finite dimensional, and there is p=p(a,b)\in\N such that [\alpha^j(a),b]=0 for |j|\ge p. For systems in this class we prove the variational principle, i.e. show that P_\alpha(H) is the supremum of the quantities h_\phi(\alpha)-\phi(H), where h_\phi(\alpha) is the Connes-Narnhofer-Thirring dynamical entropy of \alpha with respect to the \alpha-invariant state \phi. If H\in\A, and P_\alpha(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h_\phi(\alpha), and any state of finite maximal entropy is a trace.Comment: LaTeX2e, 20 page

Poisson boundaries of monoidal categories

Given a rigid C*-tensor category C with simple unit and a probability measure $\mu$ on the set of isomorphism classes of its simple objects, we define the Poisson boundary of $(C,\mu)$. This is a new C*-tensor category P, generally with nonsimple unit, together with a unitary tensor functor $\Pi: C \to P$. Our main result is that if P has simple unit (which is a condition on some classical random walk), then $\Pi$ is a universal unitary tensor functor defining the amenable dimension function on C. Corollaries of this theorem unify various results in the literature on amenability of C*-tensor categories, quantum groups, and subfactors.Comment: v2: 37 pages, minor changes, to appear in Ann. Sci. Ecole Norm. Sup.; v1: 37 page

Drinfeld center and representation theory for monoidal categories

Motivated by the relation between the Drinfeld double and central property (T) for quantum groups, given a rigid C*-tensor category C and a unitary half-braiding on an ind-object, we construct a *-representation of the fusion algebra of C. This allows us to present an alternative approach to recent results of Popa and Vaes, who defined C*-algebras of monoidal categories and introduced property (T) for them. As an example we analyze categories C of Hilbert bimodules over a II$_1$-factor. We show that in this case the Drinfeld center is monoidally equivalent to a category of Hilbert bimodules over another II$_1$-factor obtained by the Longo-Rehren construction. As an application, we obtain an alternative proof of the result of Popa and Vaes stating that property (T) for the category defined by an extremal finite index subfactor $N \subset M$ is equivalent to Popa's property (T) for the corresponding SE-inclusion of II$_1$-factors. In the last part of the paper we study M\"uger's notion of weakly monoidally Morita equivalent categories and analyze the behavior of our constructions under the equivalence of the corresponding Drinfeld centers established by Schauenburg. In particular, we prove that property (T) is invariant under weak monoidal Morita equivalence.Comment: v3: minor corrections, to appear in Comm. Math. Phys.; v2: 37 pages, with a new section; v1: 24 page