Weak Quantum Hypergroups in C*-Algebras

• Weak quantum hypergroups are C*-algebraic structures equipped with a completely positive coproduct, counit, and antilinear involution that generalize quantum groups.
• The construction utilizes finite index inclusions and tools like conditional expectations and Jones projections to establish a non-commutative convolution framework.
• This framework unifies classical quantum symmetries by interpolating between quantum hypergroups and weak Hopf algebras, extending Haar integration to non-commutative settings.

A weak quantum hypergroup is a generalization of quantum hypergroups that arises naturally from the study of finite index inclusions of simple unital C*-algebras. Formally, it consists of a unital C*-algebra $A$ equipped with a completely positive coproduct, a counit, and an antilinear involutive operation, all subject to axioms that generalize those of quantum groups and weak Hopf algebras. Unlike quantum groups and ordinary quantum hypergroups, the comultiplication in a weak quantum hypergroup need not be an algebra homomorphism, and its counit and unit possess weakened multiplicativity and comultiplicativity properties. These objects unify the frameworks of quantum hypergroups in the sense of Chapovsky–Vainerman and the weak Hopf C*-algebras of Nikshych–Vainerman, providing a canonical and C*-algebraic analytic geometry for generalized quantum symmetries (Bakshi et al., 18 Jan 2026, Delvaux et al., 2010).

1. Definition and Axioms

Given a unital C*-algebra $(A, \cdot, 1, *)$, a weak quantum hypergroup is specified by a quadruple $(A, \Delta, \varepsilon, \sharp)$, with:

• $\Delta: A \rightarrow A \otimes A$ a linear, coassociative, completely positive (CP) map,
• $\varepsilon: A \rightarrow \mathbb{C}$ a linear counit,
• $\sharp: A \rightarrow A$ an antilinear involution (i.e., $\sharp \circ \sharp = \mathrm{id}_A$).

These must satisfy the axioms:

1. Coassociativity: $$(\Delta \otimes \mathrm{id}) \circ \Delta = (\mathrm{id} \otimes \Delta) \circ \Delta$$.
2. Counitality: $$(\varepsilon \otimes \mathrm{id}) \circ \Delta = (\mathrm{id} \otimes \varepsilon) \circ \Delta = \mathrm{id}_A$$.
3. Anti-comultiplicativity: $$\Delta \circ \sharp = \Pi \circ (\sharp \otimes \sharp) \circ \Delta$$, with $\Pi(a \otimes b) = b \otimes a$.
4. Involution: $\sharp \circ \sharp = \mathrm{id}_A$.
5. Compatibility:
   • $$(uv)^\sharp = u^\sharp v^\sharp, \ \forall u,v \in A$$,
   • $\Delta \circ * = (* \otimes *) \circ \Delta$,
   • $\sharp \circ * = * \circ \sharp$.
6. Positivity: $\Delta$ is completely positive.
7. Weak Multiplicativity of the Counit:

$$\varepsilon(xyz) = \varepsilon(x y_{(1)}) \varepsilon(y_{(2)} z) = \varepsilon(x y_{(2)}) \varepsilon(y_{(1)} z) \quad \text{with} \ \Delta(y) = y_{(1)} \otimes y_{(2)}.$$

1. Weak Comultiplicativity of the Unit:

$$(\mathrm{id} \otimes \Delta) \Delta(1) = (\Delta(1) \otimes 1)(1 \otimes \Delta(1)) = (1 \otimes \Delta(1))(\Delta(1) \otimes 1).$$

If additionally $\varepsilon$ is multiplicative and $\Delta(1) = 1 \otimes 1$, the structure reduces to a quantum hypergroup in the sense of Chapovsky–Vainerman (Bakshi et al., 18 Jan 2026).

2. Construction from Finite Index C*-Inclusions

Let $B \subset A$ be an inclusion of simple unital C*-algebras with unique minimal conditional expectation $E_0: A \rightarrow B$ of finite Watatani index. The Jones tower $B \subset A \subset A_1 \subset A_2 \subset \ldots$ yields the second relative commutant $C = B' \cap A_1$. The construction on $C$ proceeds as follows:

• Fourier Transform: $F_1: C \rightarrow A' \cap A_2$ is defined by $F_1(x) = \tau^{-3/2} E_2(x v_2)$, where $v_2 = e_2 e_1$ and $\tau$ is the inverse of the minimal index.
• Convolution Product: $x \star y := F_1^{-1}(F_1(y) F_1(x))$ gives a natural non-commutative convolution structure.
• Coproduct: For $x \in C$ and $y,z \in C$, $$\langle \Delta(x), y \otimes z \rangle = \tau^{1/2} \langle x, y \star z \rangle$$ using the Markov trace $\operatorname{tr}$.
• Counit: $\varepsilon(x) = \tau^{-1} \langle e_1, x \rangle$.
• Reflection/Involution: $x^{\sharp} = r_1^+(x^*)$ with $r_1^+$ a trace-preserving involutive *-anti-homomorphism.

All axioms of a weak quantum hypergroup hold for the system $(C, \Delta, \varepsilon, \sharp)$, with the completely positive coproduct and compatibility properties verified via properties of the conditional expectations, Jones projections, and convolution product (Bakshi et al., 18 Jan 2026).

3. Structural Properties and Axiomatic Verification

The construction ensures:

• Coassociativity: $\Delta$ inherits coassociativity via associativity of convolution and trace properties.
• Counitality: Verified by explicit computation with the Markov trace.
• Anti-comultiplicativity: Utilizes that $r_1^+$ intertwines convolution with its opposite and is a *-anti-homomorphism.
• Normalization: $\sharp \circ \sharp = \mathrm{id}$ is immediate from the involutive property.
• Compatibility: Multiplicativity, *-compatibility, and commutation with $\sharp$ are direct.
• Weak Counit/Comultiplicativity: Proven using quasi-basis and Fourier-convolution computations.
• Positivity: Derived from the CP nature of the conditional expectations and *-isomorphisms linking coproduct structure to relative commutants (Bakshi et al., 18 Jan 2026).

4. Relation to Quantum Hypergroups and Weak Hopf Algebras

• Irreducible Case: If $B \subset A$ is irreducible ($B' \cap A = \mathbb{C} \cdot 1$), $\Delta(1) = 1 \otimes 1$ and $\varepsilon$ is multiplicative, so $(C, \Delta, \varepsilon, \sharp)$ reduces to the Chapovsky–Vainerman quantum hypergroup (Bakshi et al., 18 Jan 2026).
• Depth 2 Case: Inclusions $B \subset A$ of depth 2 yield $C = B' \cap A_1$ as a Nikshych–Vainerman weak Hopf C*-algebra, with its structure maps recoverable from the weak quantum hypergroup construction, up to scalar multiples. Thus, all major forms of finite quantum symmetry, including Hopf algebra, weak Hopf algebra, and quantum hypergroup, fall within this analytic weak quantum hypergroup framework (Bakshi et al., 18 Jan 2026).

5. Algebraic Quantum Hypergroups and Group-Like Projections

The notion of algebraic quantum hypergroups, often called "weak quantum hypergroups," appears naturally in algebraic settings, for instance, as function algebras on groups constant on double cosets of a finite subgroup:

• For $A = K(G)$ (finitely supported functions on a group $G$) and $H \subset G$ finite, the subalgebra $A_H$ of functions constant on $H$-double cosets admits a twisted coproduct:

$$\Delta_H(f)(p,q) = \frac{1}{|H|} \sum_{r \in H} f(prq).$$

This is coassociative and counital but, in general, not an algebra map (Delvaux et al., 2010).

• Group-like projections in a multiplier Hopf algebra give rise to general algebraic quantum hypergroups via subalgebra constructions and conditional expectations, with dual compact-type and discrete-type forms generated accordingly (Delvaux et al., 2010).

6. Haar Integrals and Invariant Measures

Weak quantum hypergroups, whether analytic or algebraic, possess a unique normalized Haar integral and invariant measure structure:

• In the analytic case, the Jones projection $e_1 \in C$ serves as a left and right Haar integral $h$, fulfilling $xh = \varepsilon^t(x) h$ and $hx = h \varepsilon^s(x)$ for derived target and source counits. The Markov trace $\varphi = \operatorname{tr}$ is a left- and right-invariant measure, satisfying $$(\mathrm{id} \otimes \varphi) \circ \Delta(x) = (\varphi \otimes \mathrm{id}) \circ \Delta(x) = \varphi(x) 1$$. These properties extend the classical Haar integration theory to the non-commutative, weakly structured setting (Bakshi et al., 18 Jan 2026).
• In the algebraic setting, restrictions of faithful integrals and counits from ambient multiplier Hopf algebras descend to the hypergroup level, preserving invariance axioms (Delvaux et al., 2010).

7. Examples and Unifying Role

Weak quantum hypergroups encompass a broad array of quantum symmetry frameworks, including:

• Double-coset (hypergroup) algebras: E.g., $B = C(K) \subset A = C(G)$ for compact $G$ and closed $K$, yielding classical hypergroups in irreducible settings.
• Depth 2 subfactors: Nikshych–Vainerman weak Hopf algebras as C*-algebraic analogues.
• Arbitrary finite-index simple C*-inclusions: Providing genuinely new structures that interpolate between quantum hypergroups and weak Hopf algebras without assuming algebraic integrability or planar-algebra data.

The weak quantum hypergroup concept thus unifies and generalizes classical, Hopf-algebraic, and operator-algebraic approaches to quantum symmetry, grounded entirely in the analytic structure of C*-algebra inclusions and their associated commutants (Bakshi et al., 18 Jan 2026, Delvaux et al., 2010).