Discrete quantum groups

Abstract: Discrete quantum groups were first introduced as the duals of compact quantum groups in a paper by Podleś and Woronowicz (1990). Later they were studied independently by Effros and Ruan (1994) and myself (1996). All of this was done before the duality of multiplier Hopf algebras with integrals was developed (1998), as a special and motivating case of the theory of locally compact quantum groups, developed even later (2000).

Unfortunately, also in more recent work, discrete quantum groups have still been treated as duals of compact quantum groups and not as a concept of its own.

In this talk I will discuss a somewhat updated version of the theory of discrete quantum groups. Given a discrete quantum group $(A,\Delta)$, I will focus on the properties of $\Delta(h)$ where $h$ is the cointegral. The element $\Delta(h)$ is a \emph{separability idempotent} in the multiplier algebra $M(A\otimes A)$ carrying all the essential information about the discrete quantum group.

I plan to use the discrete quantum group that arises from the Jimbo deformation of the enveloping algebra of the Lie algebra of $SU(2)$ to illustrate some of the notions and properties of general discrete quantum groups.

operator algebrasquantum algebra

Audience: researchers in the topic

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Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

The zoom links will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

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