Weak quantum hypergroups from finite index C*-inclusions

TL;DR

This paper introduces weak quantum hypergroups derived from finite index C*-algebra inclusions, providing a unified C*-algebraic framework for generalized quantum symmetries with concrete algebraic structures.

Contribution

It constructs a canonical coproduct on the second relative commutant and defines weak quantum hypergroups, extending quantum hypergroup theory within a C*-algebraic context.

Findings

Every finite index inclusion yields a weak quantum hypergroup.

The weak quantum hypergroup has a Haar integral.

In special cases, it recovers known structures like weak Hopf algebras.

Abstract

We study a finite index inclusion of simple unital C*-algebras and construct a canonical completely positive coproduct on the second relative commutant, thereby endowing it with a natural coalgebra structure. Motivated by this construction, we introduce the notion of a weak quantum hypergroup, a generalization of the quantum hypergroups of Chapovsky and Vainerman. We show that every finite index inclusion gives rise to such a weak quantum hypergroup, and that the corresponding weak quantum hypergroup possesses a Haar integral. In the irreducible case, this structure satisfies the axioms of a quantum hypergroup in the sense of Chapovsky and Vainerman, while in the depth 2 setting our framework yields the associated weak Hopf algebra constructed by Nikshych and Vainerman. These results provide a unified and intrinsically C*-algebraic framework for generalized quantum symmetries associated with finite index inclusions.