Algebraic quantum hypergroups and duality

Abstract: Let $G$ be a finite group and $H$ a subgroup. The set $\mathcal{G}$ of double cosets $HpH$, with $p \in G$ has the structure of an hypergroup. The product of two elements $HpH$ and $HqH$ is the set of cosets $HrH$ where $r \in pHq$. The algebra $A$ of functions on $\mathcal{G}$ is the space of functions on $G$ that are constant on double cosets. It carries a natural coproduct, dual to the product, and given by $$∆(p,q) = \frac{1}{n} \sum_{h \in H} f(phq)$$ where $n$ is the number of elements in $H$. The dual algebra is known as the Hecke algebra associated with the pair $G,H$. In this talk I will discuss the notion of an algebraic quantum hypergroup, its fundamental properties and duality for algebraic quantum hypergroups. I will illustrate this with an example, coming from bicrossproduct theory, constructed from a pair of closed subgroups $H$ and $K$ of a group $G$, with the assumption that $H \cap K = {e}$. This is part of more general work in progress with M. Landstad (NTNU, Trondheim)

operator algebrasquantum algebra

Audience: researchers in the topic

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.

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