Thomas Timmermann - Academia.edu (original) (raw)

Papers by Thomas Timmermann

$Research paper thumbnail of Partial actions of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>C</mi><mo lspace="0em" rspace="0em">∗</mo></msup></mrow><annotation encoding="application/x-tex">C^{*}</annotation></semantics></math>C∗ -quantum groups$

Banach Journal of Mathematical Analysis, Oct 1, 2018

Journal of Noncommutative Geometry, 2015

Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamica... more Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang-Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of type II 1 factors. In this article, we associate to suitable dynamical quantum groups, which are a purely algebraic objects, Hopf C˚-bimodules and measured quantum groupoids on the level of von Neumann algebras. Assuming invariant integrals on the dynamical quantum group, we first construct a fundamental unitary which yields Hopf bimodules on the level of C˚-algebras and von Neumann algebras. Next, we assume properness of the dynamical quantum group and lift the integrals to the operator algebras. In a subsequent article, this construction shall be applied to the dynamical SU q p2q studied by Koelink and Rosengren.

Journal of Algebra, Sep 1, 2015

Communications in Algebra, Sep 15, 2017

Banach Center Publications, 2015

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possib... more It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This relation has been considered already in the original paper on multiplier Hopf algebroids. In this note, we investigate the connection further. First we show that any regular weak multiplier Hopf algebra gives rise, in a natural way, to a regular multiplier Hopf algebroid. Secondly we give a characterization, mainly in terms of the base algebra, for a regular multiplier Hopf algebroid to have an underlying weak multiplier Hopf algebra. We illustrate this result with some examples. In particular, we give examples of multiplier Hopf algebroids that do not arise from a weak multiplier Hopf algebra.

International Journal of Mathematics, Feb 1, 2016

arXiv (Cornell University), Nov 3, 2009

arXiv (Cornell University), Sep 28, 2018

arXiv (Cornell University), Mar 20, 2014

A fundamental feature of quantum groups is that many come in pairs of mutually dual objects, like... more A fundamental feature of quantum groups is that many come in pairs of mutually dual objects, like finite-dimensional Hopf algebras and their duals, or quantisations of function algebras and of universal enveloping algebras of Poisson-Lie groups. The same phenomenon was studied for quantum groupoids in various settings. In the purely algebraic setup, the construction of a dual object was given by Schauenburg and by Kadison and Szlachányi, but required the quantum groupoid to be finite with respect to the base. A sophisticated duality for measured quantum groupoids was developed by Enock, Lesieur and Vallin in the setting of von Neumann algebras. We propose a purely algebraic duality theory without any finiteness assumptions, generalising Van Daele's duality theory of multiplier Hopf algebras and borrowing ideas from the theory of measured quantum groupoids. Our approach is based on the multiplier Hopf algebroids recently introduced by Van Daele and the author, and on a new approach to integration on algebraic quantum groupoids. The main concept are left and right integrals on regular multiplier Hopf algebroids that are adapted to quasi-invariant weights on the basis. Given such integrals, we show that they are unique up to rescaling, admit modular automorphisms, and that left and right ones are related by modular elements. Then, we construct, without any finiteness or Frobenius assumption, a dual multiplier Hopf algebroid with integrals and prove biduality.

$Research paper thumbnail of Free dynamical quantum groups and the dynamical quantum group \mathrmSU\mathrmdynQ(2)$

Banach Center Publications, 2012

arXiv (Cornell University), Oct 21, 2008

arXiv (Cornell University), Dec 21, 2007

arXiv (Cornell University), Mar 20, 2017

arXiv (Cornell University), 2008

arXiv (Cornell University), Jul 2, 2013

Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algeb... more Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication. We show that bijectivity of two associated canonical maps is equivalent to the existence of an antipode, discuss invertibility of the antipode, and present some examples and special cases.

arXiv (Cornell University), Sep 28, 2007

arXiv (Cornell University), Jul 28, 2009

We introduce a relative tensor product of C *-modules and a spatial fiber product of C *-algebras... more We introduce a relative tensor product of C *-modules and a spatial fiber product of C *-algebras that are analogues of Connes' fusion of correspondences and the fiber product of von Neumann algebras introduced by Sauvageot, respectively, and study their categorical properties. These constructions form the basis for our approach to quantum groupoids in the setting of C *-algebras that is published separately.

Mathematische Zeitschrift, Sep 25, 2010

A compactification of Fell is applied to locally compact non-Hausdorff groupoids and yields local... more A compactification of Fell is applied to locally compact non-Hausdorff groupoids and yields locally compact Hausdorff groupoids. In theétale case, this construction provides a geometric picture for the left-regular representations introduced by Khoshkam and Skandalis.

Advances in Mathematics, Mar 1, 2017

Journal of Noncommutative Geometry, Mar 23, 2018

We construct a maximal counterpart to the minimal quantum grouptwisted tensor product of C *-alge... more We construct a maximal counterpart to the minimal quantum grouptwisted tensor product of C *-algebras studied by Meyer, Roy and Woronowicz [16, 17], which is universal with respect to representations satisfying certain braided commutation relations. Much like the minimal one, this product yields a monoidal structure on the coactions of a quasi-triangular C *-quantum group, the horizontal composition in a bicategory of Yetter-Drinfeld C *-algebras, and coincides with a Rieffel deformation of the non-twisted tensor product in the case of group coactions.