Weak quantum hypergroups from finite index $C^*$-inclusions

Abstract: We study finite index inclusions $B \subset A$ of simple unital $C^*$-algebras and investigate the quantum symmetries arising from their relative commutants. Using the convolution structure on higher relative commutants, we construct a canonical completely positive coproduct on the second relative commutant $B^{\prime} \cap A_1$, which gives it a natural coalgebra structure. This leads to the notion of a weak quantum hypergroup. We show that such a structure arises canonically from any finite index inclusion. In the irreducible case it becomes a quantum hypergroup, while in the depth $2$ case it recovers the weak Hopf algebra associated with the inclusion. This is a joint work with  Debashish Goswami and Biplab Pal

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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