Workshop: Deligne categories and quantum groups Mar 30 – Apr 3, 2020 RWTH Aachen University, Germany organized by Johannes Flake, Gerhard Hiß, Laura Maaßen, and Moritz Weber supported by SFB TRR 195
More details. Coming soon.
Speakers. (subject to updates)
Topics. We aim at bringing together experts on Deligne's interpolation categories and their generalizations on the one hand and from representation categories of (compact) quantum groups on the other. The goal is to encourage discussions, to provide a forum for an exchange of techniques and recent developments, and to push forward interactions and transfers between these two domains. Deligne categories are constructions which interpolate the categories of representations for a given family of groups, such as general linear groups, orthogonal groups, symplectic groups, or symmetric groups. In the latter case, Deligne constructed a family of monoidal categories Rep(St), where t is any complex parameter. If t equals a positive integer n, we recover essentially the representation category of the n-th symmetric group, but in general, interesting monoidal categories are obtained in this way. Since their introduction, various properties of these interpolation categories have been determined and a number of generalizations, including to other families of groups or even families of module categories, have been considered. One crucial feature of Deligne categories is their diagrammatic description which allow for a quite combinatorial treatment. On the other hand, compact matrix quantum groups, as defined by Woronowicz in the 1980s, prove to have a rich representation theory, too – and a pretty similar diagrammatic appearance. Amongst others, Woronowicz's q-deformation SUq(2) of the special unitary group, Wang's liberated quantum version SN+ and ON+ of the symmetric and the orthogonal groups as well as Banica and Speicher's class of so called easy quantum groups obey a combinatorics which is similar to the one of Deligne categories. Moreover, modern directions in the representation theory of compact quantum groups are based on purely categorical points of views such as the one of C*-tensor categories.
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