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Weak quantum hypergroups from finite index C*-inclusions

Keshab Chandra Bakshi, Debashish Goswami, Biplab Pal

TL;DR

The work constructs a canonical completely positive coproduct on the second relative commutant $B'\cap A_1$ for a finite index inclusion $B\subset A$ of simple unital $C^*$-algebras, endowing it with a coalgebra structure and introducing weak quantum hypergroups as a natural generalization of quantum hypergroups. It proves the existence of Haar and invariant measures within this framework and shows that irreducible inclusions yield genuine quantum hypergroups, while depth-2 inclusions recover the Nikshych–Vainerman weak Hopf algebra; together, these results provide an intrinsic $C^*$-algebraic formalism for generalized quantum symmetries arising from finite index inclusions. The paper also clarifies dualities with the relative commutant $A'\cap A_2$ and connects the constructed coalgebra to existing Hopf-type objects in subfactor theory, unifying several prior approaches under a single analytic, $C^*$-algebraic umbrella.

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.

Weak quantum hypergroups from finite index C*-inclusions

TL;DR

The work constructs a canonical completely positive coproduct on the second relative commutant for a finite index inclusion of simple unital -algebras, endowing it with a coalgebra structure and introducing weak quantum hypergroups as a natural generalization of quantum hypergroups. It proves the existence of Haar and invariant measures within this framework and shows that irreducible inclusions yield genuine quantum hypergroups, while depth-2 inclusions recover the Nikshych–Vainerman weak Hopf algebra; together, these results provide an intrinsic -algebraic formalism for generalized quantum symmetries arising from finite index inclusions. The paper also clarifies dualities with the relative commutant and connects the constructed coalgebra to existing Hopf-type objects in subfactor theory, unifying several prior approaches under a single analytic, -algebraic umbrella.

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.
Paper Structure (6 sections, 29 theorems, 86 equations)

This paper contains 6 sections, 29 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inclusions of simple unital $C^*$-algebras
  4. Quantum hypergroups
  5. Convolution product and coalgebra structure
  6. Weak quantum Hypergroup structure on the second relative commutant

Key Result

Lemma 2.1

For every $x_1 \in A_1$ there exists a unique $x_0 \in A$ such that $x_1 e_1 = x_0 e_1$. In fact, $x_0 = \tau^{-1} E_1(x_1 e_1).$

Theorems & Definitions (42)

  • Lemma 2.1: BakshiVedlattice
  • Lemma 2.2: KajiwaraWatatani
  • Proposition 2.3: BakshiVedlattice
  • Theorem 2.4: BGPS
  • Definition 2.5: CV
  • Definition 2.6: CV
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Proposition 3.4
  • ...and 32 more