On free wreath products of classical groups (2512.11477v1)

Abstract: We study the generalized free wreath product of classical groups introduced by the first author and Arthur Troupel. We give an explicit computation of the Haar state and deduce important properties of their associated operator algebra: in many cases, the von Neumann algebra is a full type ${\rm II}_1$-factor and the reduced C*-algebra is simple with unique trace.

• The paper presents an explicit combinatorial formula for the Haar state using non-crossing partitions and Möbius functions.
• The paper shows that under specific group conditions, the resulting operator algebras are full II₁-factors and simple C*-algebras with a unique trace.
• The paper employs crossed product constructions to connect quantum automorphism groups with classical group dynamics, impacting quantum symmetry theory.

Summary of “On free wreath products of classical groups” (2512.11477)

Introduction and Context

The paper rigorously investigates the structure and operator algebraic properties of the generalized free wreath product of classical groups, denoted as $G = \widehat{\Gamma}\wr_{*,\beta}\Lambda$, with $\Gamma$ a discrete group and $\Lambda$ a finite group acting by left translation on $C^*(\Lambda)$ endowed with its canonical trace. This construction generalizes prior notions of free wreath products in the field of quantum automorphism groups and compact quantum groups and provides new classes of examples not encompassed by earlier frameworks.

Structure and Presentation of the Generalized Free Wreath Product

The generalized free wreath product $G$ is defined via generators-and-relations methods, with $C(G)$ constructed as the universal unital C*-algebra generated by $\nu_\gamma(g)$ for $g \in \Gamma$, $\gamma \in \Lambda$, and the group algebra $C^*(\Lambda)$, subject to specific intertwining relations that encode the quantum (and, in the classical case, group-theoretic) structure as well as compatibility between $\Lambda$ and $\Gamma$.

The quantum group structure is enforced by a comultiplication, and prior results have established connections between properties of $\Gamma$ and quantum properties of $\widehat{G}$, such as the Haagerup property, hyperlinearity, and coamenability.

Haar State and Its Explicit Combinatorial Formula

A central contribution of the paper is the explicit computation of the Haar state $h$ of $G$. The authors provide a detailed, combinatorial formula for the Haar state, which leverages non-crossing partitions, the Möbius function, and group-theoretic constraints on the family of generators. For $\vec{\gamma}\in\Lambda^n$, $\vec{g}\in\Gamma^n$, and $s\in\Lambda$:

$$h(\nu_{\vec{\gamma}}(\vec{g})s) = \delta_{s,1} \sum_{\pi \in (\vec{g}) \cap (\vec{\gamma})} \frac{\mu_\pi(\vec{g})}{|\Lambda|^{n - |\pi|}}$$

where $(\vec{g})$ and $(\vec{\gamma})$ denote the sets of partitions with respect to the product constraints in $\Gamma$ and $\Lambda$, respectively. This formula exhibits the deep interplay between combinatorial free probability (non-crossing partitions, free cumulants) and the algebraic structures at play.

The construction also ensures that $h$ is a trace and uniquely determines the Haar state in this setting.

Operator Algebras Associated to the Free Wreath Product

Crossed Product Description

The C*-algebra $C(G)$ is shown to admit a canonical crossed product description $U \rtimes_\alpha \Lambda$, where $U$ encodes the universal quantum homomorphism and $\alpha$ is the action induced by conjugation in $\Lambda$. This interplay between the universal property, representation theory, and crossed products is essential for analyzing the operator algebraic properties of $C(G)$.

von Neumann Algebra and Reduced C*-Algebra Structure

By analyzing the Haar state and the crossed product structure, the paper rigorously determines the associated von Neumann algebra $(G)$ and reduced C*-algebra $C_r(G)$. The main results are:

• If $\Gamma$ is icc and $\Lambda$ has trivial center, then $(G)$ is a full ${\rm II}_1$-factor and $C_r(G)$ is simple with unique trace.
• In the case where $\Lambda$ has nontrivial center, the center of $(G)$ is explicitly identified as ${\rm L}(\mathcal{Z}(\Lambda))$, corresponding to the group von Neumann algebra of the center.

The proofs of factoriality, fullness, and simplicity rely on a careful analysis of the action’s outerness, the universal property, the noncommutative crossed product machinery, and operator algebra techniques including applications of the structure theory of von Neumann algebras, the free product construction, and results from subfactor theory.

Implications and Directions

This work gives precise conditions under which the operator algebras arising from such free wreath products possess desirable properties—fullness in the sense of Connes/Jones, simplicity, and uniqueness of trace. This directly impacts the construction of new II$_1$ factors and simple C*-algebras arising from quantum group methods. The explicit combinatorial description of the Haar state offers a tool for further investigation of quantum invariants and their probabilistic properties.

The theoretical implications extend to the classification of quantum groups via their associated algebras, analysis of approximation properties, and connections with free probability. Practically, the framework and formulas enable explicit computation in examples, which is essential for applications to quantum invariant theory, quantum symmetry, and connections to subfactor theory.

Open directions include:

• Extending the analysis to infinite groups $\Lambda$ or to more general quantum group actions.
• Investigating approximation properties and exactness, for settings where $\Gamma$ and $\Lambda$ are not both finite.
• Deeper exploration of the representation categories and fusion rules arising from these constructions.

Conclusion

The paper presents a comprehensive operator algebraic analysis of generalized free wreath products of classical groups, producing explicit formulas for the Haar state and characterizing the associated C*- and von Neumann algebras under natural group-theoretic hypotheses. This framework deepens the interplay between combinatorial free probability, quantum group theory, and operator algebras, and lays groundwork for further developments in the structure and classification of quantum symmetries and their analytical counterparts.