Algebraic Quantum Group

• Algebraic quantum groups are regular multiplier Hopf algebras with integrals that generalize classical Hopf algebras, forming a foundation for quantum symmetries.
• They feature a robust duality theory through canonical pairings and analytic structures, linking algebraic frameworks with operator-algebraic approaches.
• Their constructions, including bicrossproducts and quantum hypergroups, underpin applications in noncommutative geometry, harmonic analysis, and quantum groupoids.

An algebraic quantum group is a regular multiplier Hopf algebra with integrals, providing a powerful generalization of Hopf algebras and forming a foundational structure for the paper of quantum symmetries, duality, and noncommutative harmonic analysis. The theory integrates non-unital algebraic frameworks, duality via pairings, analytic structures, and modular properties, and it is foundational for both the purely algebraic and the analytic/operator-algebraic approaches to quantum groups and quantum groupoids.

1. Definition and Fundamental Structures

An algebraic quantum group is defined as a pair (A, Δ), where A is an associative algebra (typically over ℂ) with nondegenerate product (often not unital), and Δ: A → M(A ⊗ A) is a coproduct mapping into the multiplier algebra of the tensor product. The coproduct Δ must be coassociative “up to multipliers” and satisfy the following conditions:

• For all a ∈ A, the elements Δ(a)(1 ⊗ b) and (a ⊗ 1)Δ(b) span A ⊗ A for b ∈ A.
• There exists a counit ε: A → ℂ, and an invertible antipode S: A → A, generalizing the Hopf algebra structure.
• A distinguishing feature is the existence of faithful (left and right) integrals; that is, linear functionals φ, ψ: A → ℂ invariant under the coproduct, e.g.,

$$(\phi \otimes \mathrm{id})\Delta(a) = \phi(a)\;1 = (\mathrm{id} \otimes \phi)\Delta(a),$$

for all $a \in A$.

Algebraic quantum groups allow for the coproduct Δ to be defined on potentially infinitely generated or nonunital algebras, with all structural maps extended to the multiplier algebras $M(A)$ or $M(A \otimes A)$, providing the flexibility to treat both finite and infinite cases (Delvaux et al., 2010, Daele, 2023).

2. Duality Theory

A core property of algebraic quantum groups is duality. For an algebraic quantum group $(A, \Delta)$, the dual $\widehat{A}$ is defined by a suitable subspace of the linear dual of $A$, typically generated by functionals of the form $a \mapsto \phi(a c)$ for $c \in A$, equipped with a coproduct and antipode defined via duality relations:

• If $(A, Δ)$ is a regular multiplier Hopf algebra with integrals, so is its dual $(\widehat{A}, \widehat{Δ})$.
• The canonical duality is implemented by the element $V \in M(\widehat{A} \otimes A)$ such that for all $a \in A$, $b \in \widehat{A}$,

$(V, a \otimes b) = \langle a, b \rangle.$

This $V$ satisfies

$$(\Delta \otimes \mathrm{id})(V) = V_{13} V_{23}, \quad (\mathrm{id} \otimes \Delta)(V) = V_{12} V_{13}, \quad V(a \otimes 1) = \Delta(a)V,$$

with corresponding pentagon relations (Daele, 2023).

• The antipode satisfies $$V^{-1} = (S \otimes \mathrm{id})(V) = (\mathrm{id} \otimes S)(V)$$, reflecting the dual relationship between algebraic operations on $A$ and coalgebraic operations on $\widehat{A}$.

This duality theory allows one to move seamlessly between $A$ and $\widehat{A}$, with structure maps and analytical constructs transferring via the canonical pairing and the element $V$.

3. Analytic and Modular Structures

The analytic and modular structure of algebraic quantum groups is deeply intertwined:

• Given a *-algebraic quantum group (i.e., with an involution * and positive integrals), the space $A$ is spanned by common eigenvectors for $S^2$, the modular automorphisms $\sigma$, $\sigma'$, and left/right multiplication by the modular element $\delta$, with all eigenvalues strictly positive (Daele, 2023).
• Positive integrals yield modular automorphism groups $\sigma$, $\sigma'$ (with $\phi(\sigma(a)b) = \phi(a\sigma'(b))$), as well as a modular element $\delta$ (a positive invertible multiplier) relating left and right integrals. The analytic structure allows definition of one-parameter groups (e.g., $a \mapsto \delta^{it} a \delta^{-it}$), providing a direct link with KMS theory and operator-algebraic quantum groups.
• The polar decomposition of the canonical operator $T$ (associated with the GNS representation of the integral) yields

$T = J \Delta^{1/2},$

connecting algebraic and analytical modular structures. The modular theory hence mirrors the Tomita–Takesaki modular theory for locally compact quantum groups and their von Neumann algebra completions.

Operator-valued quantum invariants involved in the analytic and modular structures directly control the scaling, modular automorphisms, and weight theory, and allow passage to full locally compact quantum group frameworks upon completion.

4. Bicrossproducts and Additional Constructions

Algebraic quantum groups admit rich constructions via bicrossproducts and related operations:

• The bicrossproduct $A \# B$ of algebraic quantum groups $A$ and $B$ (each a regular multiplier Hopf algebra with integrals) is formed when $B$ is a right $A$-module algebra and $A$ is a left $B$-comodule coalgebra, with matched data. The twisted product structure and coproducts encode complex symmetry features such as "quantum doubles" and nontrivial modular data (Delvaux et al., 2012). For example:

$$\Delta_{\#}(a \# b) = (a_{(1)} \# 1) \cdot \Gamma(a_{(2)}) \cdot (1 \# b),$$

with modular elements, modular automorphisms, and scaling determined from those of the factors and an extra multiplier $y \in M(B)$.

• The duality theory for bicrossproducts ensures that the dual of a bicrossproduct is again a bicrossproduct of the duals, a key structural persistence when passing between algebraic and dual analytic settings.

Integrals, scaling constants, and modular data for the bicrossproduct depend intricately on the corresponding data for the constituent quantum groups and the compatibility maps.

5. Examples and Applications

Algebraic quantum groups encompass a spectrum of structures:

• The algebra $K(G)$ of finitely supported functions on a (possibly infinite) group $G$, with coproduct $\Delta(f)(p,q) = f(pq)$, is a canonical algebraic quantum group, generalizing the classical Hopf algebra of a finite group (Delvaux et al., 2010).
• Duality with group algebras and richer examples—matched pairs of groups yield bicrossproduct algebraic quantum groups that generalize both group algebras and function algebras on groups.
• Generalizations to quantum groupoids (weak multiplier Hopf algebras with integrals) extend the theory to cases with local units and multiple "bases" (Daele, 2017).

Applications lie in noncommutative geometry, representation theory, categorical frameworks, as well as in the analytic construction of locally compact and operator-algebraic quantum groups (see (Daele, 2023) for detailed modular and analytic links).

6. Algebraic Quantum Hypergroups and Beyond

Relaxing the requirement that the coproduct be multiplicative leads to algebraic quantum hypergroups, further extending the reach of the theory:

• In algebraic quantum hypergroups, the coproduct is assumed only linear and coassociative (with $\Delta(1) = 1 \otimes 1$), preserving the existence of integrals and antipodes but embracing more general, “averaged” symmetry structures (Delvaux et al., 2010, Landstad et al., 2022).
• Quantum hypergroups naturally arise by restriction/averaging procedures (e.g., averaging over subgroups or via group-like projections), and the duality framework extends to these structures, at least in the finite case.
• Algebraic quantum groups thus provide a natural backbone for the more general theory of quantum symmetries, accommodating both rigid and hypergroupical deformations.

7. Impact and Further Developments

Algebraic quantum groups unify the algebraic, analytic, and representation-theoretic threads of quantum symmetry:

• The duality, analytic, and modular structures provide machinery for constructing noncommutative harmonic analysis, spectral theory, and operator-algebraic quantum groups (Daele, 2023, Daele, 2023).
• Their bicrossproduct and duality frameworks underlie physically relevant examples including quantum doubles, quantizations of Poisson–Lie groups, and quantum transformation groupoids (Kahng, 2011, Delvaux et al., 2012, Taipe, 2023).
• Applications extend to the categorification of quantum groups, K-theoretic invariants, and the realization of quantum symmetries in mathematical physics (Qin, 2013, Ellis, 2014).

Algebraic quantum groups also serve as a crucial bridge between the algebraic and operator-algebraic approaches—every algebraic quantum group with positive integrals yields, upon completion, a locally compact quantum group in the sense of Kustermans–Vaes, while many analytic properties can be traced to their algebraic origins (Daele, 2023). This integrative role is central to the ongoing development of quantum group theory in mathematics and mathematical physics.