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Quantization of locally compact groups associated with essentially bijective $1$-cocycles

This paper constructs a dual unitary 2-cocycle on a locally compact group extension to define a cocycle bicrossed product deformation of its dual, generalizing previous finite group results and providing a quantization framework for solutions to the Yang-Baxter equation and brace structures via irreducible projective representations and Kohn-Nirenberg type maps.

Original authors: Pierre Bieliavsky, Victor Gayral, Sergey Neshveyev, Lars Tuset

Published 2026-08-27

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CC BY 4.0

Original authors: Pierre Bieliavsky, Victor Gayral, Sergey Neshveyev, Lars Tuset

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a field dedicated to understanding symmetry, not just in the shapes we see, but in the invisible structures that govern how things combine and transform. This is the study of groups, which are essentially collections of objects that can be put together in specific ways, much like how musical notes form chords or how puzzle pieces fit together. For decades, mathematicians have been fascinated by a particular challenge: how to take these classical, predictable structures and "quantize" them. This process involves bending the rigid rules of classical symmetry to fit the strange, probabilistic logic of quantum mechanics, where things can exist in multiple states at once. The goal is to create new mathematical objects called quantum groups, which act as bridges between the smooth, continuous world of classical physics and the discrete, jittery world of the very small. To do this, researchers often look for specific patterns within these groups that allow them to be reshaped without breaking their fundamental nature.

A team of mathematicians has now expanded the toolkit for building these quantum bridges, moving beyond the specific, well-trodden paths they had explored in previous work. Their new paper tackles a broad class of mathematical structures known as locally compact groups, which include both finite collections of elements and continuous, flowing systems like the real number line. The researchers focused on a specific setup where a large group is built by extending a smaller, simpler group with an abelian one—a structure that can be visualized as a complex machine built upon a stable, predictable foundation. The key to their success was finding a way to map the elements of one part of this machine onto the dual of another part in a way that is essentially one-to-one, covering the space efficiently without gaps or overlaps. They call this an "essentially bijective" connection. By ensuring this connection holds true, they were able to construct a new kind of mathematical object that deforms the original group into a quantum version.

The researchers achieved this by inventing a new way to represent the group's actions on a space of functions, creating a specific type of projection that preserves the group's internal logic while allowing it to be reshaped. This method allowed them to define a precise rule, or "cocycle," that governs how the quantum version of the group behaves. Unlike earlier approaches that worked only for very specific, rigid cases or required the groups to be finite, this new construction works for a much wider variety of systems, including those that are infinite and continuous. The result is a family of new quantum groups that are mathematically sound and structurally rich. These new objects are not just abstract curiosities; they are defined by a specific relationship between two subgroups that fit together perfectly, a concept known in mathematics as a "matched pair."

What makes this discovery particularly significant is its connection to some of the most famous puzzles in theoretical physics and mathematics. The researchers showed that their method applies to every "brace" structure, a type of algebraic system that arises naturally when solving the Yang–Baxter equation, a fundamental formula that describes how particles interact and scatter in quantum systems. In simpler terms, the paper proves that if you have a solution to this famous equation that behaves in a certain non-degenerate way, you can automatically generate a corresponding quantum group. This unifies several previously separate lines of inquiry, showing that the work of earlier researchers on finite groups and the authors' own previous work on specific continuous groups are actually special cases of a single, more general principle.

The paper also clarifies the relationship between these new quantum groups and the mathematical "cohomology" that classifies them. Cohomology can be thought of as a way to count the different ways a structure can be twisted or deformed. The authors demonstrated that when their specific mapping is a simple, direct transformation, there is a one-to-one correspondence between the possible twists in the original group and the resulting quantum groups. This answers a lingering question from their earlier research, confirming that the method generates a complete and distinct family of solutions in these cases. However, they also noted that when the mapping is more complex, this perfect one-to-one relationship can break down, meaning some potential quantum groups might be missed or some might look the same even if they came from different starting points.

Ultimately, this work provides a robust, general framework for turning classical symmetry groups into quantum ones. It does not rely on the groups being finite or having a specific geometric shape, but rather on the existence of a specific, efficient mapping between their components. By constructing a new type of representation and a corresponding quantization map, the authors have shown how to systematically generate these quantum objects. The result is a clearer, more comprehensive picture of how classical symmetries can be deformed into their quantum counterparts, offering a powerful new lens through which to view the mathematical structures underlying the quantum world. The findings are presented as rigorous proofs, establishing a solid foundation for future exploration in the theory of quantum groups and their applications to the Yang–Baxter equation.

Technical Summary of "Quantization of Locally Compact Groups Associated with Essentially Bijective 1-Cocycles"

Problem Statement
The paper addresses the construction of locally compact quantum groups via the deformation of the dual of a locally compact group $G$ . Specifically, given an extension of locally compact groups $0 \to V \to G \to Q \to 1$ where $V$ is abelian, the authors seek to construct a unitary dual 2-cocycle $\Omega$ on $G$ . Such a cocycle satisfies the identity $(\Omega \otimes 1)(\hat{\Delta} \otimes \iota)(\Omega) = (1 \otimes \Omega)(\iota \otimes \hat{\Delta})(\Omega)$ and defines a new coproduct $\hat{\Delta}_\Omega = \Omega \hat{\Delta}(\cdot) \Omega^*$ , yielding a locally compact quantum group $\hat{G}_\Omega$ .

This work generalizes previous settings:

The "dual orbit condition" introduced in the authors' earlier work [6, 8], which applies to specific Lie groups and requires a measure class isomorphism between $Q$ and an orbit in $\hat{V}$ .
The finite group constructions by Etingof and Gelaki [9] and Ben David and Ginosar [5], which utilize bijective 1-cocycles.

The central challenge is to extend the finite group framework of bijective 1-cocycles to the analytical setting of locally compact groups, where bijectivity is replaced by "essential bijectivity" (measure class isomorphism), and to handle general extensions (not just semidirect products) defined by a 2-cocycle $\beta \in Z^2(Q; V)$ .

Methodology
The authors employ a strategy based on the construction of a specific irreducible projective representation and a corresponding quantization map, following the approach of De Commer [7] for infinite groups.

Setup and Compatibility:
- Let $\eta: Q \to \hat{V}$ be a continuous 1-cocycle. It is termed essentially bijective if the map is a measure class isomorphism.
- Let $\beta \in Z^2(Q; V)$ define the extension $G$ .
- A compatibility condition is imposed: the 3-cocycle $\psi \in Z^3(Q; \mathbb{T})$ defined by $\psi(q_1, q_2, q_3) = e^{i\langle \eta(q_1^{-1}), \beta(q_2, q_3) \rangle}$ must be trivial in cohomology ( $[\psi] = [1]$ ). This ensures the existence of a 2-cochain $b$ such that $\psi = \partial b$ .
Projective Representation:
- A 2-cocycle $\omega$ on $G$ is constructed using $\eta$ , $\beta$ , and $b$ .
- An irreducible, square-integrable projective unitary representation $\pi: G \to PU(L^2(Q))$ is defined explicitly. The action involves the character $\eta$ , the cochain $b$ , and the modular function of $Q$ .
- The pair $(B(L^2(Q)), \text{Ad } \pi)$ is shown to be a $G$ -Galois object. This relies on proving the ergodicity and integrability of the action and establishing the unitarity of the associated Galois map.
Quantization Map:
- A unitary operator $\text{Op}: L^2(G) \to \text{HS}(L^2(Q))$ is constructed. This map intertwines the left regular representation of $G$ with the adjoint action $\text{Ad } \pi$ .
- The construction is of the Kohn–Nirenberg type, utilizing a partial Fourier transform on the abelian subgroup $V$ and the unitary isomorphism induced by the essentially bijective cocycle $\eta$ .
Dual Cocycle and Multiplicative Unitary:
- Using the Galois object and the quantization map, an explicit formula for the dual unitary 2-cocycle $\Omega$ is derived.
- The multiplicative unitary $\hat{W}_\Omega$ of the deformed quantum group is computed. It is expressed as a composition of a pentagonal transformation $v$ on $(Q \times \hat{V}) \times (Q \times \hat{V})$ and a multiplication operator by a pentagonal 2-cocycle $\Theta$ .

Key Results

Identification of the Quantum Group: The deformed quantum group $\hat{G}_\Omega$ is identified as a cocycle bicrossed product. Specifically, it is isomorphic to the quantum group defined by the matched pair of subgroups $(Q, Q_\eta)$ within the semidirect product $Q \rtimes \hat{V}$ , where $Q_\eta = \{(q, \eta(q)) \mid q \in Q\}$ .
- This yields a short exact sequence of locally compact quantum groups: $1 \to \hat{Q} \to \hat{G}_\Omega \to Q_\eta \to 1$ .
Explicit Formulas: The paper provides explicit formulas for:
- The dual 2-cocycle $\Omega$ in terms of the Fourier transform.
- The multiplicative unitary $\hat{W}_\Omega$ in terms of the pentagonal transformation $v$ and the cocycle $\Theta$ .
- The transformation $v$ and cocycle $\Theta$ are defined using $\eta$ , the cochain $b$ , and the 3-cocycle $\psi$ .
Classification via Kac Cohomology:
- The construction yields a family of dual cocycles parameterized by the cohomology group $H^2(Q; \mathbb{T})$ .
- The authors analyze the map $H^2(Q; \mathbb{T}) \to H^2_{\text{pent}}(v; \mathbb{T})$ (where the target is the pentagonal cohomology of the matched pair).
- Injectivity Result: If $\eta$ is a coboundary (i.e., $\eta(q) = q \cdot \xi_0 - \xi_0$ ), the map from $H^2(Q; \mathbb{T})$ to the pentagonal cohomology is injective. This answers a question left open in the authors' previous work [8].
- If $\eta$ is not a coboundary, the map is not necessarily injective; the kernel is related to the restriction of cocycles on $Q \rtimes \hat{V}$ to $Q$ and $Q_\eta$ .

Significance and Scope
The paper claims to significantly expand the class of groups admitting such quantizations beyond the dual orbit condition and finite groups.

Generalization: It unifies the finite group case (Etingof-Gelaki, Ben David-Ginosar) and the Lie group case (dual orbit condition) under the framework of essentially bijective 1-cocycles and compatible extensions.
New Structures: The construction generates locally compact quantum groups from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation and, more generally, from every brace structure (via Example 1.6).
Technical Contribution: The paper provides a streamlined verification of the Galois property that does not rely on deformations of function algebras, instead utilizing the explicit construction of the projective representation and the quantization map.
Geometric Interpretation: For Lie groups, essential bijectivity corresponds to the existence of a $Q$ -invariant affine structure on $Q$ .

The work does not propose new experimental applications but rather establishes a rigorous analytical framework for constructing specific classes of locally compact quantum groups, resolving open questions regarding the parameterization of these deformations in terms of group cohomology.

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