[Paper Review] Free wreath product quantum groups : the monoidal category, approximation properties and free probability

François Lemeux, Pierre Tarrago|arXiv (Cornell University)|Nov 15, 2014

Advanced Operator Algebra Research48 references18 citations

TL;DR

This paper establishes the fusion rules for free wreath product quantum groups $γ\wr_*S_N^+$ for compact matrix quantum groups $γ$ of Kac type and $N \geq 4$, using combinatorial descriptions of intertwiner spaces. It proves monoidal equivalence between $γ\wr_*S_N^+$ and a quantum group with dual in the free product $\widehat{\gamma}*\widehat{SU_q(2)}$, leading to stability results for operator algebras such as the Haagerup property, weak amenability, and exactness.

ABSTRACT

In this paper, we find the fusion rules for the free wreath product quantum groups $\\mathbb{G}\\wr_*S_N^+$ for all compact matrix quantum groups of Kac type $\\mathbb{G}$ and $N\\ge4$. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of $\\mathbb{G}\\wr_*S_N^+$. The combinatorial properties of the intertwiner spaces in $\\mathbb{G}\\wr_*S_N^+$ then allows us to obtain several probabilistic applications. We then prove the monoidal equivalence between $\\mathbb{G}\\wr_*S_N^+$ and a compact quantum group whose dual is a discrete quantum subgroup of the free product $\\widehat{\\mathbb{G}}*\\widehat{SU_q(2)}$, for some $0<q\\le1$. We obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as Haagerup property, weak amenability and exactness.

Motivation & Objective

• To determine the fusion rules for free wreath product quantum groups $γ\wr_*S_N^+$ when $γ$ is a compact matrix quantum group of Kac type and $N \geq 4$.
• To establish a monoidal equivalence between $γ\wr_*S_N^+$ and a compact quantum group whose dual is a discrete quantum subgroup of $\widehat{\gamma}*\widehat{SU_q(2)}$ for $0 < q \leq 1$.
• To derive stability results for operator algebras associated with free wreath products, including the Haagerup property, weak amenability, and exactness.
• To apply the Tannaka-Krein duality and combinatorial intertwiner space structures to analyze probabilistic and algebraic properties of the quantum groups.

Proposed method

• Use of combinatorial descriptions of intertwiner spaces between generating representations of $γ\wr_*S_N^+$ to derive fusion rules.
• Application of Tannaka-Krein duality to relate corepresentations and $C^*$-tensor categories of the quantum group.
• Construction of monoidal equivalence via the dual of a quantum group embedded in $\widehat{\gamma}*\widehat{SU_q(2)}$.
• Use of recursive relations and algebraic identities involving $A_L(X)$ and $d_\alpha$ to compute dimensions of irreducible corepresentations.
• Leveraging known results on $S_N^+$ and $SU_q(2)$, particularly Brannan’s work on the Haagerup property, to extend operator algebraic properties.
• Employment of induction and case analysis on the structure of corepresentations $\rho = b^{l_1}\alpha_1\dots b^{l_k}$ to prove dimension formulas.

Experimental results

Research questions

• RQ1 What are the fusion rules for the free wreath product quantum group $γ\wr_*S_N^+$ when $γ$ is of Kac type and $N \geq 4$?
• RQ2 Is there a monoidal equivalence between $γ\wr_*S_N^+$ and a quantum group whose dual lies in the free product $\widehat{\gamma}*\widehat{SU_q(2)}$?
• RQ3 How do the intertwiner spaces of $γ\wr_*S_N^+$ enable probabilistic and algebraic applications?
• RQ4 What operator algebraic properties—such as the Haagerup property, weak amenability, and exactness—are preserved under free wreath product construction?
• RQ5 Can the dimension of an irreducible corepresentation of $γ\wr_*S_N^+$ be computed via a product formula involving $A_L(\sqrt{N})$ and quantum dimensions $d_\alpha$?

Key findings

• The fusion rules for $γ\wr_*S_N^+$ are fully determined via a combinatorial description of intertwiner spaces between generating representations.
• The dimension of an irreducible corepresentation $ρ = b^{l_1}\alpha_1\dots b^{l_k}$ is given by $\text{dim}(\rho) = \prod_{i=1}^{k-1} d_{\alpha_i} \prod_{i=1}^k A_{l_i}(\sqrt{N})$.
• A monoidal equivalence is established between $γ\wr_*S_N^+$ and a compact quantum group whose dual is a discrete quantum subgroup of $\widehat{\gamma}*\widehat{SU_q(2)}$ for some $0 < q \leq 1$.
• As a consequence, the reduced $C^*$-algebra and von Neumann algebra of $γ\wr_*S_N^+$ inherit the Haagerup property, weak amenability, and exactness from the target quantum group.
• The intertwiner space structure enables probabilistic applications, including results on free compound Poisson variables, extending prior work on $\widehat{\mathbb{Z}/s\mathbb{Z}}\wr_*S_N^+$.
• The proof relies on recursive decomposition of corepresentations using tensor product identities and induction on the number of $b$-factors, with case distinctions based on the length of the last block.