Locally Compact Quantum Group

• Locally compact quantum groups are noncommutative analogs of classical groups defined via C*-algebras and Haar weights.
• They integrate duality, convolution structures, and subgroup lattices to extend harmonic analysis and quantum probability frameworks.
• The operator-algebraic approach fosters studies on approximation properties like the Haagerup property and property (T), influencing quantum symmetry research.

A locally compact quantum group is a non-commutative generalization of a locally compact group, formulated within the framework of operator algebras. It serves as a foundational object in non-commutative harmonic analysis, quantum probability, and the structure theory of quantum symmetries. The concept unifies algebraic, analytical, and categorical properties that extend the duality, cohomology, representations, and subgroup structures of classical groups into the non-commutative, operator algebraic regime.

1. Abstract Structure and Fundamental Definitions

A locally compact quantum group (LCQG) is formalized as a pair (A, Δ), where A is a C*-algebra or von Neumann algebra and Δ: A → M(A ⊗ A) is a comultiplication, i.e., a non-degenerate, coassociative -homomorphism: $$(\Delta \otimes \mathrm{id}) \Delta = (\mathrm{id} \otimes \Delta) \Delta.$$ This structure is supplemented by left and right Haar weights—n.s.f. weights φ, ψ satisfying invariance conditions analogous to Haar measures on classical groups. The framework encompasses both reduced (C₀(G)) and universal (C₀^u(G)) C-algebraic quantum groups, von Neumann algebraic quantum groups (where A is replaced by M), and dual objects (e.g., Fourier transforms and dual quantum groups).

Key structural elements include:

• The multiplicative unitary W ∈ M(A ⊗ A) that implements the comultiplication:

$\Delta(x) = W^* (1 \otimes x) W$

• The scaling group (Tₜ), antipode S, and modular element δ, generalizing the modular function and inversion in locally compact groups.
• Haar weights φ and ψ (left and right, respectively), providing the quantum analog of Haar measure.

The operator algebra setting (C*-algebraic/von Neumann algebraic) facilitates the integration of analytical techniques in non-commutative geometry and quantum probability.

2. Duality and Convolution Structures

Intrinsic duality is a defining feature: To each LCQG (A, Δ), a dual object (Â, Δ̂) is constructed, itself a locally compact quantum group. Duality is made concrete via the multiplicative unitary, inducing left and right fundamental unitaries (W, V). This duality endows the trace class operators T(L²(G)) with two naturally induced products corresponding to convolution (*), and pointwise multiplication (·): $$p * q = T_*(p \otimes q), \quad p \cdot q = \mathcal{T}_*(p \otimes q).$$ These products are connected by anti-commutation relations (e.g., Theorem 3.6 in (Kalantar et al., 2011)), consolidating convolution and Fourier algebra structures within a unified operator-algebraic framework.

Convolution semigroups of states (μₜ) are governed by the comultiplication and have generating functionals γ whose domains are dense unital *-subalgebras with core-like properties. Every symmetric generating functional γ corresponds to a Dirichlet form Q on L²(G), producing Markov semigroups and quantum Lévy processes (Skalski et al., 2019).

3. Subgroups, Idempotents, and Lattice Structures

A wealth of subgroup concepts translate to the quantum setting:

• Open quantum subgroups H < G are characterized by normal, surjective, unital *-homomorphisms π: L∞(G) → L∞(H) intertwining Δ (i.e., (π ⊗ π)∘Δ^G = Δ^H∘π), and correspond bijectively to central group-like projections P ∈ Z(L∞(G)) satisfying Δ^G(P)(1 ⊗ P) = P ⊗ P (Kalantar et al., 2015, Faal et al., 2017).
• Idempotent states ω on quantum group algebras (ω * ω = ω) generalize idempotent measures on groups; they correspond to right-invariant expected C*-subalgebras and group-like projections on the dual quantum group (Salmi et al., 2011, Kasprzak et al., 2018). Haar idempotents are those arising as Haar states on compact quantum subgroups.
• Lattices of idempotent states mimic the intersection and join operations of compact subgroups, giving a noncommutative lattice structure (Kasprzak et al., 2018).

Normal integrable coideals invariant under the scaling group are in one-to-one correspondence with compact quantum subgroups (Faal et al., 2017). The center and inner automorphisms have canonical operator-algebraic descriptions, and the canonical central exact sequence

$$1 \to \mathscr{Z}(G) \to G \to \mathrm{Inn}(G) \to 1,$$

with $\mathrm{Inn}(G) \cong G/\mathscr{Z}(G)$, generalizes the classical correspondence between the center, quotient, and inner automorphism group (Kasprzak et al., 2015).

4. Cohomological, Geometric, and Harmonic Analysis Properties

Cohomological properties of module structures over convolution algebras serve as indicators of global topological properties:

• Projectivity of the trivial module over the convolution algebra characterizes compactness.
• Projectivity of L¹(G) as a right module characterizes discreteness.
• The Radon–Nikodym property of L¹(G) links to the existence of normal conditional expectations and is finely distinguished from discreteness in quantum settings (Kalantar et al., 2011). For compact Kac algebras, RNP of L¹(G) is equivalent to G being finite.

Poisson boundaries over LCQGs generalize the classical harmonic function theory. The Choquet–Deny theorem (triviality of the Poisson boundary) and Kaimanovich–Vershik/Rosenblatt amenability results are shown to have precise quantum analogs: amenability is equivalent to the existence of measures with trivial Poisson boundary (Kalantar et al., 2011).

Quantum harmonic analysis introduces operator-valued convolutions and mixed-state localization operators, relying on admissibility conditions and the Duflo–Moore theorem, thereby generalizing classical signal analysis to non-unimodular, quantum structured phase spaces (Halvdansson, 2022).

5. Rigidity, Approximation Properties, and Representation Theory

Rigidity properties and approximation criteria receive quantum generalizations:

• Property (T) is formulated for LCQGs in terms of isolation of trivial representations of the universal dual C*-algebra, spectral decomposition, and bicrossed product constructions for nonclassical Property (T) quantum groups (Chen et al., 2015, Brannan et al., 2019).
• Haagerup property is characterized via the density of mixing representations, convolution semigroups of states, existence of symmetric proper conditionally negative generating functionals, and real proper cocycles. For unimodular quantum groups, the Haagerup property is a von Neumann algebraic property preserved under free products (Daws et al., 2013).
• Similarity properties: The conjectured Day–Dixmier property (every completely bounded representation is similar to a *-representation) fails for general non-Kac type quantum groups but holds for amenable Kac type LCQGs. Explicit similarity degree bounds are obtained in this framework (Brannan et al., 2017).
• Covariant Stone–von Neumann theorem: For regular LCQGs, every Heisenberg representation of a dynamical system (G, A, α) is a multiple of the canonical Schrödinger representation on Hilbert modules, providing a new criterion for strong regularity (Hall et al., 2023).

Multiplicative unitary techniques, modular theory, and a robust duality dictate harmonic and representation-theoretic analysis, with phenomena such as maximal actions, modular representations, and the classification of crossed product and Bohr compactification structures (Daws, 2013).

6. Quantum Groupoids, Bornological Quantum Groups, and Extensions

The formalism encompassing locally compact quantum groupoids extends the group case by admitting a nontrivial base C*-algebra, a canonical idempotent E∈M(A⊗A), and adapted density and invariance axioms. The quantum group case is recovered when the base is trivial (B=ℂ), thus embedding quantum group theory into a broader groupoid framework. Duality constructions and examples (fields of quantum groups, transformation groupoids) are formulated via measured and locally compact groupoid analogs (Kahng et al., 2017, Enock, 2019).

Bornological quantum groups, as introduced by Voigt, provide dense *-subalgebras with the full analytic and Hopf structure necessary for constructing locally compact quantum groups: regular representations, modular automorphism groups, and group-like elements are concretely realized on these cores, simplifying the theory of quantum subgroups and morphisms (Rivet et al., 2021).

7. Applications, Examples, and Research Directions

LCQGs underpin a broad spectrum of applications:

• In quantum probability, idempotent states and convolution semigroups model quantum analogues of random walks, invariants, and limit distributions.
• In representation theory, the extension of property (T), Haagerup property, and uniqueness theorems to noncommutative symmetries informs rigidity theory and harmonic analysis on quantum spaces.
• Detailed constructions of dual 2-cocycles and deformations yield new LCQGs, particularly via quantization of subgroups (e.g., of the affine group over local fields), with explicit cocycle and kernel formulas (Jondreville, 2018).
• The quantum Bohr compactification yields compact quantum approximants by "cutting down" to the algebra generated by finite-dimensional corepresentations, exhibiting rich behavior in cocommutative and non-amenable cases (Daws, 2013).

The ongoing development includes refinement of structural correspondences (lattices, coideals, homogeneous spaces), the extension of classical theorems to quantum settings, and the creation of analytic and algebraic tools for classifying, deforming, and representing locally compact quantum symmetries in operator-algebraic settings.