arXiv subjects
Publications and source records attributed to K. De Commer.
We construct the first examples of purely continuous, $q$-deformed Lie type locally compact quantum groups in higher rank. They arise from Drinfeld-Jimbo quantization, at unimodular deformation parameter, of the totally positive part of higher rank split real Lie groups in type $A$. Our techniques are based on quantum cluster theory, in particular as developed through the work of Fock and Goncharov.
Given a locally compact quantum group $\mathbb{G}$ and a (generalized) dual unitary $2$-cocycle $\hat{\Omega}$, any W$^*$-algebra $A$ with a $\mathbb{G}$-action can be twisted into a new W$^*$-algebra $A_{\hat{\Omega}}$ with an action by the cocycle twist $\mathbb{G}_{\hat{\Omega}}$ of $\mathbb{G}$. We show how, in general, the standard space $L^2(A_{\hat{\Omega}})$, with its standard $\mathbb{G}_{\hat{\Omega}}$-representation, can be seen as a twist of $L^2(A)$ with its standard $\mathbb{G}$-representation. We then apply this general result in the special case of (generalized) Drinfeld doubles.
Let $\mathbb{G}$ be a locally compact quantum group, and $A,B$ von Neumann algebras on which $\mathbb{G}$ acts. We refer to these as $\mathbb{G}$-dynamical W$^*$-algebras. We make a study of $\mathbb{G}$-equivariant $A$-$B$-correspondences, that is, Hilbert spaces $\mathcal{H}$ with an $A$-$B$-bimodule structure by $*$-preserving normal maps, and equipped with a unitary representation of $\mathbb{G}$ which is equivariant with respect to the above bimodule structure. Such structures are a Hilbert space version of the theory of $\mathbb{G}$-equivariant Hilbert C$^*$-bimodules. We show that there is a well-defined Fell topology on equivariant correspondences, and use this to formulate approximation properties for them. Within this formalism, we then characterize amenability of the action of a locally compact group on a von Neumann algebra, using recent results due to Bearden and Crann. We further consider natural operations on equivariant correspondences such as taking opposites, composites and crossed products, and examine the continuity of these operations with respect to the Fell topology.
Let $\mathbb{G}$ be a compact quantum group and $A\subseteq B$ an inclusion of $\sigma$-finite $\mathbb{G}$-dynamical von Neumann algebras. We prove that the $\mathbb{G}$-inclusion $A\subseteq B$ is strongly equivariantly amenable if and only if it is equivariantly amenable, using techniques from the theory of non-commutative $L^p$-spaces. In particular, if $(A, \alpha)$ is a $\mathbb{G}$-dynamical von Neumann algebra with $A$ $\sigma$-finite, the action $\alpha: A \curvearrowleft \mathbb{G}$ is strongly (inner) amenable if and only if the action $\alpha: A \curvearrowleft \mathbb{G}$ is (inner) amenable. By duality, we also obtain the same result for $\mathbb{G}$ a discrete quantum group, so that, in particular, a discrete quantum group is inner amenable if and only it is strongly inner amenable. This result can be seen as a dynamical generalization of Tomatsu's result on the amenability/co-amenability duality. We also provide the first explicit examples of amenable discrete quantum groups that act non-amenably on a von Neumann algebra.
We show that the duals of Woronowicz's quantum SU(2)-groups converge, within the operator algebraic setting, to the group of special upper triangular 2-by-2 matrices with positive diagonal.
We show that the family of Podles' spheres is complete under equivariant Morita equivalence (with respect to the action of quantum SU(2)), and determine the associated orbits. We also give explicit formulas for the actions which are equivariantly Morita equivalent with the quantum projective plane. In both cases, the computations are made by examining the localized spectral decomposition of a generalized Casimir element.
In this article, we investigate the notion of a Galois object for a locally compact quantum group M. Such an object consists of a von Neumann algebra N equipped with an ergodic integrable coaction of M on N, such that the crossed product is a type I factor. We show how to construct from such a coaction a new locally compact quantum group P, which we call the reflection of M along N. By way of application, we prove the following statement: any twisting of a locally compact quantum group by a unitary 2-cocycle is again a locally compact quantum group.
Let SU_q(2) and E_q(2) be Woronowicz' q-deformations of respectively the compact Lie group SU(2) and the non-trivial double cover of the Lie group E(2) of Euclidian transformations of the plane. We prove that, in some sense, their duals are `Morita equivalent locally compact quantum groups'. In more concrete terms, we prove that the von Neumann algebraic quantum groups of `bounded measurable functions' on SU_q(2) and E_q(2) are unitary cocycle deformations of each other.
In this note we construct bi-*-Galois objects linking the quantized universal enveloping algebras associated to the Lie groups SU(2), E(2) and SU(1,1), where E(2) denotes the Lie group of Euclidian transformations of the plane, and we show how one can create (formal) quantum homogeneous spaces for these quantum groups by integrating the associated Miyashita-Ulbrich action on certain subquotient *-algebras.
We supplement the study of Galois theory for algebraic quantum groups started in the paper 'Galois Theory for Multiplier Hopf Algebras with Integrals' by A. Van Daele and Y.H. Zhang. We examine the structure of the Galois objects: algebras equipped with a Galois coaction such that only the scalars are coinvariants. We show how their structure is as rich as the one of the quantum groups themselves: there are two distinguished weak K.M.S. functionals, related by a modular element, and there is an analogue of the antipode squared. We also show how to reflect the quantum group across the Galois object to obtain a (possibly) new algebraic quantum group. We end by considering an example.
Let $(A,Δ)$ be a locally compact quantum group and $(A_0,Δ_0)$ a regular multiplier Hopf algebra. We show that if $(A_0,Δ_0)$ can in some sense be imbedded in $(A,Δ)$, then $A_0$ will inherit some of the analytic structure of $A$. Under certain conditions on the imbedding, we will be able to conclude that $(A_0,Δ_0)$ is actually an algebraic quantum group with a full analytic structure. The techniques used to show this, can be applied to obtain the analytic structure of a $^*$-algebraic quantum group {\it in a purely algebraic fashion}. Moreover, the {\it reason} that this analytic structure exists at all, is that the one-parameter groups, such as the modular group and the scaling group, are diagonizable. In particular, we will show that necessarily the scaling constant $μ$ of a $^*$-algebraic quantum group equals 1. This solves an open problem.