相关论文: Free product formulae for quantum permutation grou…

A free wreath product construction of a Hopf algebra (or of a Woronowicz algebra) by Wang's quantum permutation group is done. It provides new examples of quantum groups and is useful to describe the quantum automorphism group of the…

We introduce the free inhomogeneous wreath product of compact matrix quantum groups, which generalizes the free wreath product (Bichon 2004). We use this to present a general technique to determine quantum automorphism groups of connected…

Let $\mathbb{G}$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,\psi)$ and $\Gamma$ a discrete group. We want to compute the fusion rules of $\widehat{\Gamma}\wr_* \mathbb{G}$. First of all, we will revise the…

Associated to a discrete group $G$, one has the topological category of finite dimensional (unitary) $G$-representations and (unitary) isomorphisms. Block sums provide this category with a permutative structure, and the associated…

Let $\mathbb{G}$ be a compact quantum group and $\mathbb{G}^{aut}(B,\psi)$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,\psi)$. In this paper, we study the free wreath product $\mathbb{G}\wr_{*}…

Consider the wreath product $\Gamma = F\wr \mathrm{F_n} = \bigoplus_{\mathrm{F_n}}F\rtimes\mathrm{F_n}$, with $F$ a finite group and $\mathrm{F_n}$ the free group on $n$ generators. We study the Baum-Connes conjecture for this group. Our…

By a construction of Vaughan Jones, the bipartite graph $\Gamma(A)$ associated with the natural inclusion of $\mathbb C$ inside a finite-dimensional $C^*$-algebra $A$ gives rise to a planar algebra $\mathcal P^{\Gamma(A)}$. We prove that…

The automorphism group of the composition of graphs $G \circ H$ contains the wreath product $Aut(H) \wr Aut(G)$ of the automorphism groups of the corresponding graphs. The classical problem considered by Sabidussi and Hemminger was under…

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…

If $G$ is a free product of finite groups, let $\Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {\em…

We give a characterisation of quantum automorphism groups of trees. In particular, for every tree, we show how to iteratively construct its quantum automorphism group using free products and free wreath products. This can be considered a…

In this note we observe that any compact quantum group monoidally equivalent, in a nice way, to a free wreath product of a compact quantum group $G$ by the quantum automorphism group of a finite dimensional C*-algebra with a $\delta$-form…

We study the free product of rooted graphs and its various decompositions using quantum probabilistic methods. We show that the free product of rooted graphs is canonically associated with free independence, which completes the proof of the…

Sabidussi's theorem [Duke Math. J. 28, 1961] gives necessary and sufficient conditions under which the automorphism group of a lexicographic product of two graphs is a wreath product of the respective automorphism groups. We prove a quantum…

We define a free product of connected simple graphs that is equivalent to several existing definitions when the graphs are vertex-transitive but differs otherwise. The new definition is designed for the automorphism group of the free…

The notion of the quantum automorphism group of a graph was introduced by J. Bichon in 2003 and T. Banica in 2005 respectively. This article explores primarily the quantum automorphism group of a graph $\Gamma$, denoted by…

We investigate a construction which associates a finite von Neumann algebra $M(\Gamma,\mu)$ to a finite weighted graph $(\Gamma,\mu)$. Pleasantly, but not surprisingly, the von Neumann algebra associated to to a `flower with $n$ petals' is…

In this paper we find the fusion rules of the free wreath products $\widehat{\Gamma}\wr_*S_N^+$ for any (discrete) group $\Gamma$. To do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify…

We develop a new approach on free wreath products, generalizing the constructions of Bichon and of Fima-Pittau. We show stability properties for certain approximation properties such as exactness, Haagerup property, hyperlinearity and…

Let $G$ and $H$ be finitely generated groups. In this paper, we prove the quantitative coarse Baum--Connes conjecture for the free product $G* H$ under the assumption that the conjecture holds for both $G$ and $H$.