Multiplier Hopf Algebras with Integrals

• Multiplier Hopf algebras with integrals are non-unital algebras equipped with a coassociative coproduct and nonzero invariant functionals, extending classical Hopf algebra theory.
• Faithful left and right integrals ensure the existence of a unique counit and antipode, underpinning duality, modularity, and completeness in the algebraic structure.
• Twist deformations and analytic completions of these algebras bridge the gap to locally compact quantum groups and enhance modern quantum group analysis.

A multiplier Hopf algebra with integrals is a (possibly nonunital) algebra equipped with a coassociative coproduct in the multiplier algebra and admitting nonzero left and right invariant functionals, called left and right integrals. The existence and properties of such integrals extend the classical finite-dimensional Hopf algebra theory to the infinite-dimensional and non-unital setting, and they provide the structural foundation for duality, modularity, and analytic completion to locally compact quantum groups. The theory is closely related to that of weak multiplier Hopf algebras and multiplier Hopf algebroids, and it underpins modern quantum group analysis.

1. Structure and Definitions

Let $A$ be a non-unital but nondegenerate algebra over a field $\Bbbk$, with the multiplier algebra $M(A)$. A regular multiplier Hopf algebra is a pair $(A, \Delta)$ with an algebra homomorphism

$\Delta: A \to M(A \otimes A)$

satisfying coassociativity: $$(\Delta \otimes \mathrm{id})\Delta = (\mathrm{id} \otimes \Delta)\Delta \quad \text{as maps}\quad A \to M(A \otimes A \otimes A).$$ Canonical maps $T_1(a \otimes b) = \Delta(a)(1 \otimes b)$ and $T_2(a \otimes b) = (a \otimes 1)\Delta(b)$ are required to be bijections, and regularity further demands all four canonical maps determined by leg placements to be bijective.

A counit $\varepsilon:A\to\Bbbk$ and antipode $S:A\to M(A)$ are uniquely determined by the triangular equations

$$(\varepsilon\otimes\mathrm{id})\bigl(\Delta(a)(1\otimes b)\bigr) = ab,\quad m(S\otimes\mathrm{id})\bigl(\Delta(a)(1\otimes b)\bigr)=\varepsilon(a)b,$$

and analogously on the other leg.

A left integral is a nonzero linear form $\phi:A\to\Bbbk$ such that $(\mathrm{id}\otimes\phi)\Delta(a) = \phi(a)1$; a right integral $\psi$ satisfies $(\psi\otimes\mathrm{id})\Delta(a) = \psi(a)1$. Integrals are unique up to scalar, and when they exist, $(A, \Delta)$ is called an algebraic quantum group (Yang, 2015).

2. Integrals: Existence, Uniqueness, and Faithfulness

Faithful integrals play a decisive role in the structure theory. Faithfulness of a linear functional $\omega$ means $\omega(ab) = 0~\forall~b\implies a=0$ and $\omega(ba) = 0~\forall~b\implies a=0$. For a regular multiplier Hopf algebra, the existence of both a faithful left and right integral guarantees fullness ($$\Delta(A)(1\otimes A) = (A\otimes1)\Delta(A) = A\otimes A$$), the bijectivity of canonical maps, and the existence of a counit and antipode, thus providing a full Hopf algebra structure in the multiplier context (Daele, 2024).

Uniqueness of integrals up to scalar follows by invariance properties and nondegeneracy of the product (Yang, 2015). In the algebroid setting, full integrals are unique up to multiplication by central multipliers in the base algebras (Timmermann, 2014).

3. Twisting and Deformation of Integrals

The Drinfelʹd twist construction generalizes to multiplier Hopf algebras. Given an invertible, normalized twist $J \in M(A \otimes A)$ satisfying the cocycle condition and normalization, the twisted comultiplication

$\Delta^J(a) = J^{-1} \Delta(a) J$

defines a new multiplier Hopf algebra structure. The integrals deform via the invertible multiplier $u_J = Q_J S(Q_J)$, with

$$\phi^J(a) = \phi(u_J a), \quad \psi^J(a) = \psi(a u_J^{-1}),$$

where $Q_J = m(S \otimes \mathrm{id})(J)$ and $S^J(a) = Q_J^{-1} S(a) Q_J$ is the twisted antipode. These twisted integrals remain left and right invariant for $(A, \Delta^J)$, ensuring that Drinfelʹd twist preserves the algebraic quantum group property and faithfulness of integrals (Yang, 2015).

4. Duality and Modular Structure

Multiplier Hopf algebras with integrals admit a canonical dual structure. The dual Hopf algebra consists of functionals of the form $\phi(\cdot\, a)$, with convolution as multiplication. The pairing is non-degenerate and dualizes the Hopf algebra structure, enabling biduality: the original algebra is recovered as the dual of the dual (Timmermann, 2014).

Integrals determine a modular element $\delta \in M(A)$ that implements the square of the antipode $S^2(a) = \delta a \delta^{-1}$, and modular automorphism $\sigma$ with $\phi(ab) = \phi(b \sigma(a))$. These modular objects generalize the Tomita–Takesaki theory for von Neumann algebras and are crucial for analytic aspects (Timmermann, 2014).

For *-algebras with positive integrals, the associated Hilbert space constructions (via the GNS representation) and the associated unitary representations can be completed to analytic quantum groups, culminating in the full locally compact quantum group structure (Daele, 2023).

5. Extensions: Weak Multiplier Hopf Algebras and Algebroids

The framework extends to weak multiplier Hopf algebras, where the existence of a faithful set of integrals implies the presence of a counit and an antipode, forming a regular weak multiplier Hopf algebra. In this setting, the canonical idempotent $E \in M(A \otimes A)$ and partial source and target subalgebras play a central role, and the integral theory enables lifting the bialgebra structure to a full Hopf algebra structure. Faithful cointegrals also provide sufficient and necessary conditions for integrals to exist (Kahng et al., 2014, Zhou et al., 2017).

Multiplier Hopf algebroids generalize this structure to multiple base algebras, where integrals are bimodule maps and modularity and uniqueness extend naturally (Timmermann, 2014). The duals of these algebroids are constructed using integrals, and biduality recovers the original algebraic object without finiteness assumptions.

6. Examples and Analytic Completions

Standard examples include convolution algebras of discrete groups, function algebras on compact groups, groupoid algebras, and infinite-dimensional Hopf quasigroups and their integral duals. These settings illustrate the various aspects of integrals, duality, and modularity in concrete terms (Yang, 2020, Daele, 2023).

In the setting of positive integrals, one obtains a direct bridge from algebraic quantum groups to the theory of locally compact quantum groups, as positive integrals induce Haar weights and the multiplicative unitaries needed for the analytic structure, including the modular theory and operator algebraic completions (Daele, 2023).

7. Role in Quantum Group Theory and Current Directions

Multiplier Hopf algebras with integrals subsume classical Hopf algebras, non-unital infinite-dimensional cases, and operator-algebraic quantum groups. They provide a unifying algebraic framework for duality, modularity, and analytic quantum group theory. The theory facilitates the study of locally compact quantum groupoids, extension to weak and measured settings, Drinfelʹd-type deformations, and categorical duality (Yang, 2015, Timmermann, 2014, Daele, 2024). Current research continues to refine the relationships between integrals, modular structures, and analytic completions, with applications across noncommutative geometry, quantum groupoids, and infinite-dimensional representation theory.