Reduced Free Unitary Compact Quantum Groups

• Reduced free unitary compact quantum groups are C*-algebraic quantum groups defined by universal unitary relations, completed via the Haar state, and characterized by free fusion semirings.
• They exhibit robust analytic properties, including the Haagerup and metric approximation properties, which ensure faithful representations and enable finite-dimensional approximations.
• Their noncommutative fusion rules and simplicity, confirmed by quantum analogues of Kurosh’s theorem, underscore their central role in noncommutative geometry and quantum probability.

A reduced free unitary compact quantum group is a specific analytic realization of the universal “free unitary” quantum group, constructed as the completion of the dense -algebra of matrix coefficients under the norm induced by the Haar state, resulting in a C-algebraic quantum group with noncommutative, “free” representation theory. These objects underpin a paradigmatic family in noncommutative geometry, quantum probability, and modern operator algebra theory, and serve as models for quantum symmetries beyond the classical Lie context.

1. Algebraic and Analytic Foundations

Let $Q$ be a positive definite matrix in %%%%1%%%%. The universal free unitary quantum group algebra, denoted %%%%2%%%%, is the universal unital *-algebra generated by $n^2$ elements $u_{ij}$ with relations enforcing that the matrix $u = (u_{ij})$ and its $Q$-twisted adjoint $Q\overline{u}Q^{-1}$ are both unitary; that is,

$$u u^* = u^* u = I, \qquad Q u^* Q^{-1} (Q u Q^{-1})^* = I.$$

The *-algebra $A_u(Q)$ also admits the structure maps of a compact quantum group in the sense of Woronowicz, i.e., comultiplication $\Delta(u_{ij}) = \sum_{k} u_{ik} \otimes u_{kj}$, counit $\epsilon(u_{ij}) = \delta_{ij}$, and antipode $S(u_{ij}) = u_{ji}^*$.

The reduced free unitary compact quantum group arises as the completion of $A_u(Q)$ in the GNS norm induced by the Haar state $h$, yielding the reduced C*-algebra $C_r(U_n^+)$. The universal property ensures that any unital -homomorphism from $A_u(Q)$ into a C-algebra with a matrix of unitaries subject to the above relations factors uniquely through $C_r(U_n^+)$.

A distinctive feature of the reduced completion is that the Haar state becomes faithful, enabling a robust analytic framework encompassing representation theory, functional analysis, and quantum probability (Banica, 2018, Banica, 2019).

2. Representation Theory and Fusion Semiring

The representation theory of reduced free unitary compact quantum groups is governed by a free fusion semiring structure. The simple (irreducible) corepresentations are indexed by words in two symbols, typically identified with the fundamental representation $u$ and its conjugate $\overline{u}$. The tensor product decompositions are dictated by a “free” fusion rule: $r_x r_y = \sum_{x = ag,\, y = g^* b} r_{ab},$ where $r_x$ and $r_y$ correspond to words $x$ and $y$ and $*$ denotes the involution (interchanging $u$ and $\overline{u}$). This inherently noncommutative rule gives rise to a fusion semiring which, upon passing to the quotient by the center (corresponding to balancing the number of $u$ and $\overline{u}$), structures the entire fusion algebra for the reduced (i.e., projective) version (Chirvasitu, 2012, Chirvasitu, 2012).

Explicitly, after factoring out the center (a one-dimensional torus subsystem), simple corepresentations are labeled by “balanced” words—words with equal numbers of $u$ and $\overline{u}$ symbols—leading to a projectively simple compact quantum group. This fusion semiring is noncommutative: order of tensoring matters, a stark contrast to the commutative fusion rings of classical groups.

3. Approximation Properties and Operator Algebras

The reduced C*-algebra and von Neumann algebra of a reduced free unitary compact quantum group display robust analytic approximation properties. Specifically:

• The reduced von Neumann algebra $L^\infty(U_n^+)$ has the Haagerup approximation property (HAP): there exists a net of normal, unital, completely positive, Haar-state-preserving maps with compact L^2-extensions converging strongly to the identity.
• The reduced C*-algebra $C_r(U_n^+)$ exhibits the metric approximation property (MAP): it admits a net of finite rank contractions approximating the identity in the norm topology.

These properties are established via explicit construction of approximating completely positive maps (using convolution and Haar-averaging), and verified using a Haagerup-type inequality: $$\|x\|_{C_r(U_n^+)} \leq D (n + 1) \|x\|_{L^2(U_n^+)}$$ for $x$ in the homogeneous subspace associated to representations of length $n$, with a dimension-dependent constant $D$ (Brannan, 2011).

These results are significant: while $C_r(U_n^+)$ is non-nuclear (unlike the commutative case), the existence of MAP indicates a form of finite-dimensional approximation, paralleling properties of free group factors.

4. Actions, Algebraic Cores, and Minimal Reduction

When a compact quantum group $G = (A, \Delta)$ acts on a unital C*-algebra $B$ via a -homomorphism $\delta : B \rightarrow B \otimes_{min} A$ satisfying suitable coassociativity and density conditions (the Podleś condition), two natural dense *-subalgebras arise: the **Podleś algebraic core* $B_{alg}$ (spanned by elements transforming under irreducibles), and the subalgebra $$\mathcal{B} = \{ b \in B \mid \delta(b) \in B_{alg} \otimes \mathcal{A} \}$$, with $\mathcal{A}$ the dense Hopf *-algebra in $A$.

A fundamental result is that $\mathcal{B} = B_{alg} + \ker \delta$. If $\delta$ is non-injective, the minimal reduction procedure replaces $B$ by $B_{min} = B/\ker \delta$, pushing the action to $\overline{\delta} = (p \otimes id) \circ \delta$, and ensuring injectivity without altering the algebraic core. This is particularly relevant for situations where nontrivial kernel arises naturally (as in non-reduced vs. reduced versions, and canonical actions by coproducts) (Sołtan, 2010).

5. Simplicity, Centers, and Cocenter Reduction

Reduced free unitary compact quantum groups (as in the projectivized quotient) are simple: they possess no nontrivial normal quantum subgroups beyond scalars and their own algebra. The simplicity results from quotienting by the center, which is formalized as the universal central quantum subgroup (implemented via a Hopf algebra homomorphism acting by scalars on irreducibles).

The central quotient (the cocenter) of $A_u(Q)$ maintains a fusion semiring of balanced words and, by Theorem 3.1, if the fusion semiring is free, the cocenter is simple (Chirvasitu, 2012, Chirvasitu, 2012). Furthermore, coproducts (free products) of such quantum groups remain simple after cocenter reduction under nondegeneracy conditions, vastly enlarging the family of “building block” simple quantum groups.

6. Representation-Theoretic and Asymptotic Invariants

Fusion algebras of reduced free unitary groups admit explicit computation of asymptotic invariants such as uniform Fôlner constants, the Kazhdan constant for the regular representation, and the exponential growth rate $\omega(R, d)$ (where $R$ is the fusion algebra and $d$ the dimension function). For the fusion algebra of $U_F^+$ (a standard reduced free unitary quantum group), the growth rate is the unique real root $r_q>1$ of a cubic polynomial in terms of the quantum dimension parameter $q$: $$P_q(z) = z^3 - (2q^{-2} + 3 + 2q^2)z^2 + 2(q^{-2} + 1 + q^2)z - 2.$$ These invariants reflect strong non-amenability and rapid exponential growth, distinguishing the analytic and combinatorial structure markedly from classical or amenable quantum groups. Explicit relationships such as

$$\text{F\o l}^\text{inn}(R, d) \leq 1 - 1/\omega(R, d), \qquad K(\rho, R)^2 \leq \text{F\o l}(R, d)$$

are established, encoding the lack of “approximate” invariance and indicating the depth of quantum noncommutativity (Krajczok et al., 19 Feb 2025).

7. Classification and Quantum Analogues of Kurosh's Theorem

A comprehensive combinatorial classification of all discrete quantum subgroups of $U_n^+$ has been achieved using the theory of projective partitions: every dual quantum subgroup corresponds to a standard module of projective partitions closed under concatenation and rotation. This fibered correspondence provides a quantum analogue of Kurosh’s theorem, asserting that every discrete quantum subgroup of $U_n^+$ has a “free” fusion semiring, though not every free fusion semiring arises from such a subgroup (Freslon et al., 3 Mar 2024).

Key structural families—including those arising from iterated free wreath products and free complexifications—are realized as quantum automorphism groups of quantum graphs (such as rooted regular quantum trees), providing geometric representations of the partition-theoretic structure.

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In summary, reduced free unitary compact quantum groups are operator-algebraic and representation-theoretic objects defined via universal unitary relations, completed in the GNS norm. Their fusion theory is free and noncommutative, leading to simplicity and exponential representation growth. The analytic structure admits refined approximation properties (HAP and MAP) and sharp non-amenability, with explicit combinatorial and geometric classifications for quantum subgroups. These properties make them central to noncommutative harmonic analysis, quantum symmetries, and the ongoing paper of quantum group actions on C*-algebras, with implications for noncommutative geometry and quantum probability.