Quantum Symmetries of Graph C*-Algebras

TL;DR

This paper computes the quantum automorphism groups of finite graphs and shows they act maximally on the associated graph C*-algebras, revealing that quantum symmetries are preserved in the algebraic structure.

Contribution

It establishes that the quantum automorphism group of a finite directed graph acts maximally on its graph C*-algebra, linking graph symmetries with algebraic automorphisms.

Findings

Quantum automorphism groups act maximally on graph C*-algebras.

Quantum symmetries of graphs coincide with those of their C*-algebras.

Comparison of different definitions of quantum automorphism groups for small graphs.

Abstract

The study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

Equations

$$\mathrm{Aut}(\Gamma ):=\{\sigma \in S_n\mid \sigma \epsilon =\epsilon \sigma \}\subseteq S_n,$$

$$C($$

$$=C^{\ast }(u_{ij},i,j=1,\dots ,n\mid u_{ij}=u_{ij}^{\ast }=u_{ij}^2,\sum _l{u}_{il}=1=\sum _l{u}_{lj},R_{\mathrm{Ban}}\text{ hold}),$$

$$\sum _k{u}_{ik}{\epsilon }_{kj}=\sum _k{\epsilon }_{ik}{u}_{kj}.$$

$$C($$

$$:=C^{\ast }(u_{ij},i,j=1,\dots ,n\mid u_{ij}=u_{ij}^{\ast }=u_{ij}^2,\sum _l{u}_{il}=1=\sum _l{u}_{lj},R_{\mathrm{Bic}}\text{ hold}),$$

$$\sum _k{u}_{ik}{\epsilon }_{kj}=\sum _k{\epsilon }_{ik}{u}_{kj},u_{s(e)s(f)}{u}_{r(e)r(f)}=u_{r(e)r(f)}{u}_{s(e)s(f)}\text{ for }e,f\in E,$$

$$\mathrm{Aut}(\Gamma )\subseteq G_{aut}^{\ast }(\Gamma )\subseteq G_{aut}^{+}(\Gamma )$$

$$C(G_{aut}^{+}(\Gamma ))$$

$$u_{ij}$$

$$C^{\ast }(\Gamma ):=C^{\ast }(p_v,v\in V,s_e,e\in E\mid$$

$$s_e^{\ast }{s}_e=p_{r(e)},\sum _{\begin{array}{c}e\in E\\ s(e)=v\end{array}}{s}_e{s}_e^{\ast }=p_v,\text{ if }{s}^{-1}(v)\ne \varnothing ).$$

$$\Gamma \mapsto G_{aut}^{+}(\Gamma )$$

$$\alpha :C^{\ast }(\Gamma )$$

$$p_i$$

$$s_{e_j}$$

$$\beta :C^{\ast }(\Gamma )$$

$$p_i$$

$$s_{e_j}$$

$$\mathrm{Aut}(\Gamma )=\{\sigma \in S_n|\sigma \epsilon =\epsilon \sigma \}\subseteq S_n$$

$$C(S_n^{+}):=C^{\ast }(u_{ij}|u_{ij}=u_{ij}^{\ast }=u_{ij}^2,\sum _{l=1}^n{u}_{il}=1=\sum _{l=1}^n{u}_{li}\text{ for all 1≤i,j≤n}).$$

$$C^{\ast }(p_1,\dots ,p_n|p_i=p_i^{\ast }=p_i^2,\sum _l{p}_l=1)$$

$$u_{ij}=u_{ij}^{\ast },u_{ij}{u}_{ik}={\delta }_{jk}{u}_{ij},u_{ji}{u}_{ki}={\delta }_{jk}{u}_{ji},$$

$$\sum _{l=1}^n{u}_{il}=1=\sum _{l=1}^n{u}_{li},$$

$$u_{s(e_j)i}{u}_{r(e_j)k}=u_{r(e_j)k}{u}_{s(e_j)i}=0,$$

$$u_{is(e_j)}{u}_{kr(e_j)}=u_{kr(e_j)}{u}_{is(e_j)}=0,$$

$$u_{s(e_j)s(e_l)}{u}_{r(e_j)r(e_l)}=u_{r(e_j)r(e_l)}{u}_{s(e_j)s(e_l)},$$

$$\sum _{l=1}^m{u}_{s(e_l)s(e_j)}{u}_{r(e_l)r(e_j)}=1=\sum _{l=1}^m{u}_{s(e_j)s(e_l)}{u}_{r(e_j)r(e_l)},$$

$$\sum _{l=1}^m{u}_{s(e_l)s(e_j)}{u}_{r(e_l)r(e_j)}=\sum _{i,k=1}^n{u}_{is(e_j)}{u}_{kr(e_j)}=\left(\sum _{i=1}^n{u}_{is(e_j)}\right)\left(\sum _{k=1}^n{u}_{kr(e_j)}\right)=1$$

$$u\epsilon =\epsilon u,$$

$$C(G_{aut}^{+}(\Gamma ))=C^{\ast }(u_{ij}|\text{Relations eqrefQA1 – eqrefQA4})$$

$$\mathrm{Aut}(\Gamma )\subseteq G_{aut}^{\ast }(\Gamma )\subseteq G_{aut}^{+}(\Gamma )$$

$$C(S_n^{+})=C^{\ast }(u_{ij}|\text{Relations eqrefQA1 and eqrefQA2})$$

$$\mathrm{Aut}(\Gamma )=G_{aut}^{\ast }(\Gamma )=S_n,G_{aut}^{+}(\Gamma )=S_n^{+}$$

$$\mathrm{Aut}({\Gamma }^c)=S_n,G_{aut}^{\ast }({\Gamma }^c)=G_{aut}^{+}({\Gamma }^c)=S_n^{+}$$

$$\mathrm{Aut}(\Gamma )=\mathrm{Aut}({\Gamma }^c)$$

$$G_{aut}^{+}(\Gamma )=G_{aut}^{+}({\Gamma }^c)$$

Full text

The second author was partially funded by the ERC Advanced Grant NCDFP, held by Roland Speicher. This work was part of the first author’s Master’s thesis. This work was also supported by the DFG project Quantenautomorphismen von Graphen

Introduction

Symmetry constitutes one of the most important properties of a graph. It is captured by its automorphism group

[TABLE]

where $\Gamma=(V,E)$ is a finite graph with $n$ vertices and no multiple edges, $\varepsilon\in M_{n}(\{0,1\})$ is its adjacency matrix, and $S_{n}$ is the symmetric group. In modern mathematics, notably in operator algebras, symmetries are no longer described only by groups, but by quantum groups. In 2005, Banica [1] gave a definition of a quantum automorphism group of a finite graph within Woronowicz’s theory of compact matrix quantum groups [19]. In our notation, $G_{aut}^{+}(\Gamma)$ is based on the $C^{*}$-algebra

[TABLE]

where $S_{n}^{+}$ is Wang’s quantum symmetric group [17] and $R_{\mathrm{Ban}}$ are the relations

[TABLE]

Earlier, in 2003, Bichon [5] defined a quantum automorphism group $G_{aut}^{*}(\Gamma)$ via

[TABLE]

where $R_{\mathrm{Bic}}$ are the relations

[TABLE]

and $r:E\to V$ and $s:E\to V$ are range and source maps respectively. We immediately see that

[TABLE]

holds, in the sense that there are surjective ∗-homomorphisms:

[TABLE]

Relatively little is known about these two quantum automorphism groups of graphs and we refer to Section 3.4 for an overview on all published articles in this area.

Graph $C^{*}$-algebras in turn are well-established objects in operator algebras. They emerged from Cuntz and Krieger’s work [8] in the 1980’s and they developed to be one of the most important classes of examples of $C^{*}$-algebras, see for instance Raeburn’s book for an overview [14]. Given a finite graph $\Gamma=(V,E)$ the associated graph $C^{*}$-algebra $C^{*}(\Gamma)$ is defined as

[TABLE]

A natural question is then: What is the quantum symmetry group of the graph $C^{*}$-algebra and is it one of the above two quantum automorphism groups of the underlying graphs? The answer is: It is given by the one defined by Banica. Note however, that Bichon’s definition has its justification in other contexts such as in [4, 6] or in the recent work by Speicher and the second author [15]. Moreover, Bichon’s work [5] inspired us how to formulate our main theorem, see also Remark 4.2.

1. Main result

Intuitively speaking, our main result is that the quantum symmetry of a finite graph without multiple edges coincides with the quantum symmetry of the associated graph $C^{*}$-algebra. In other words, the following diagram is commutative:

[TABLE]

More precisely, we have the following result.

Main Theorem**.**

Let $\Gamma$ be a finite graph with $n$ vertices $V=\{1,...,n\}$ and $m$ edges $E=\{e_{1},...,e_{m}\}$ having no multiple edges. The maps

[TABLE]

define a left and a right action of $G_{aut}^{+}(\Gamma)$ on $C^{*}(\Gamma)$, respectively. Moreover, whenever $G$ is a compact matrix quantum group acting on $C^{*}(\Gamma)$ in the above way, we have $G\subseteq G_{aut}^{+}(\Gamma)$. In this sense, the quantum automorphism group $G_{aut}^{+}(\Gamma)$ of $\Gamma$ is the quantum symmetry group of $C^{*}(\Gamma)$, see also Remark 4.1.

We also provide some tools for comparing and dealing with the two definitions of quantum automorphism groups of graphs, $G_{aut}^{+}(\Gamma)$ and $G_{aut}^{*}(\Gamma)$, notably depending on the complement $\Gamma^{c}$ of $\Gamma$, see Section 3.5. Moreover, we provide a list of all $\operatorname{Aut}(\Gamma)$, $G_{aut}^{+}(\Gamma)$ and $G_{aut}^{*}(\Gamma)$ for undirected graphs $\Gamma$ on four vertices, having no multiple edges and no loops, see Section 3.6.

2. Preliminaries

2.1. Graphs

We fix some notations for graphs used throughout this article. A graph $\Gamma=(V,E)$ is finite, if the set $V$ of vertices and the set $E$ of edges are finite. We denote by $r:E\to V$ the range map and by $s:E\to V$ the source map. A graph is undirected if for every $e\in E$ there is a $f\in E$ with $s(f)=r(e)$ and $r(f)=s(e)$; it is directed otherwise. Elements $e\in E$ with $s(e)=r(e)$ are called loops. A graph without multiple edges is a directed graph, where there are no $e,f\in E$, $e\neq f$, such that $s(e)=s(f)$ and $r(e)=r(f)$. For a finite graph $\Gamma=(V,E)$ with $V=\{1,\ldots,n\}$, its adjacency matrix $\varepsilon\in M_{n}(\mathbb{N}_{0})$ is defined as $\varepsilon_{ij}:=\#\{e\in E\;|\;s(e)=i,r(e)=j\}$. Here $\mathbb{N}_{0}=\{0,1,2,\ldots\}$. Throughout this article we restrict to finite graphs having no multiple edges.

If $\Gamma=(V,E)$ is a directed graph without multiple edges, we denote by $\Gamma^{c}=(V,E^{\prime})$ the complement of $\Gamma$, where $E^{\prime}=(V\times V)\backslash E$. Within the category of graphs having no loops, the complement $\Gamma^{c}$ is defined using $E^{\prime}=(V\times V)\backslash(E\cup\{(i,i);\ i\in V\})$.

2.2. Automorphism groups of graphs

For a finite graph $\Gamma=(V,E)$ without multiple edges, a graph automorphism is a bijective map $\sigma:V\to V$ such that $(\sigma(i),\sigma(j))\in E$ if and only if $(i,j)\in E$. In other words, $\varepsilon_{\sigma(i)\sigma(j)}=1$ if and only if $\varepsilon_{ij}=1$. The set of all graph automorphisms of $\Gamma$ forms a group, the automorphism group $\operatorname{Aut}(\Gamma)$. We can view $\operatorname{Aut}(\Gamma)$ as a subgroup of the symmetric group $S_{n}$, if $\Gamma$ has $n$ vertices:

[TABLE]

2.3. Graph $C^{*}$-algebras

The theory of Graph $C^{*}$-algebras has its roots in Cuntz and Krieger’s work [8] in 1980. Nowadays, it forms a well-developed and very active part of the theory of $C^{*}$-algebras, see [14] for an overview or [9] for recent developments. For a finite, directed graph $\Gamma=(V,E)$ without multiple edges, the graph $C^{*}$-algebra $C^{*}(\Gamma)$ is the universal $C^{*}$-algebra generated by mutually orthogonal projections $p_{v}$, $v\in V$ and partial isometries $s_{e}$, $e\in E$ such that

• (i)

$s_{e}^{*}s_{e}=p_{r(e)}$ for all $e\in E$

• (ii)

and $p_{v}=\sum_{e\in E\;:\;s(e)=v}s_{e}s_{e}^{*}$ for every $v\in V$ with $s^{-1}(v)\neq\emptyset$.

It follows immediately, that $s_{e}^{*}s_{f}=0$ for $e\neq f$ and $\sum_{v\in V}p_{v}=1$ hold true in $C^{*}(\Gamma)$.

2.4. Compact matrix quantum groups

Compact matrix quantum groups were defined by Woronowicz [18, 19] in 1987. They form a special class of compact quantum groups, see [12, 16] for recent books. A compact matrix quantum group $G$ is a pair $(C(G),u)$, where $C(G)$ is a unital (not necessarily commutative) $C^{*}$-algebra which is generated by $u_{ij}$, $1\leq i,j\leq n$, the entries of a matrix $u\in M_{n}(C(G))$. Moreover, the *-homomorphism $\Delta:C(G)\to C(G)\otimes C(G)$, $u_{ij}\mapsto\sum_{k=1}^{n}u_{ik}\otimes u_{kj}$ must exist, and $u$ and its transpose $u^{t}$ must be invertible.

Example 2.1**.**

As an example, consider the quantum symmetric group $S_{n}^{+}=(C(S_{n}^{+}),u)$ as defined by Wang [17] in 1998. It is the compact matrix quantum group given by

[TABLE]

One can show that the quotient of $C(S_{n}^{+})$ by the relations that all $u_{ij}$ commute is exactly $C(S_{n})$. Moreover, the symmetric group $S_{n}$ may be viewed as a compact matrix quantum group $S_{n}=(C(S_{n}),u)$, where $u_{ij}:S_{n}\to\mathbb{C}$ are the evaluation maps of the matrix entries. This justifies the name “quantum symmetric group”.

If $G=(C(G),u)$ and $H=(C(H),v)$ are compact matrix quantum groups with $u\in M_{n}(C(G))$ and $v\in M_{n}(C(H))$, we say that $G$ is a compact matrix quantum subgroup of $H$, if there is a surjective *-homomorphism from $C(H)$ to $C(G)$ mapping generators to generators. In this case we write $G\subseteq H$. As an example: $S_{n}\subseteq S_{n}^{+}$. The compact matrix quantum groups $G$ and $H$ are equal as compact matrix quantum groups, writing $G=H$, if we have $G\subseteq H$ and $H\subseteq G$.

2.5. Actions of quantum groups

Let $G=(C(G),u)$ be a compact matrix quantum group and let $B$ be a $C^{*}$-algebra. A left action of $G$ on $B$ is a unital *-homomorphism $\alpha:B\to C(G)\otimes B$ such that

• (i)

$(\Delta\otimes\mathrm{id})\circ\alpha=(\mathrm{id}\otimes\alpha)\circ\alpha$

• (ii)

and $\alpha(B)(C(G)\otimes 1)$ is linearly dense in $C(G)\otimes B$.

A right action is a unital *-homomorphism $\beta:B\to C(G)\otimes B$ with

• (i)

$((F\circ\Delta)\otimes\mathrm{id}))\circ\beta=(\mathrm{id}\otimes\beta)\circ\beta$

• (ii)

and $\beta(B)(C(G)\otimes 1)$ is linearly dense in $C(G)\otimes B$,

where $F$ is the flip map $F:C(G)\otimes C(G)\to C(G)\otimes C(G)$, $a\otimes b\mapsto b\otimes a$. Note that in some articles (for instance in [17]), the property (ii) is replaced by

• (ii’)

$(\varepsilon\otimes\mathrm{id})\circ\alpha=\mathrm{id}$

• (iii’)

and there is a dense *-subalgebra of $B$, such that $\alpha$ restricts to a right coaction of the Hopf *-algebra on the *-subalgebra.

One can show that (ii’) and (iii’) are equivalent to (ii), see [13].

2.6. Quantum symmetry group of $n$ points

According to Wang’s work [17], we know that $S_{n}^{+}$ (from Example 2.1) is the quantum symmetry group of $n$ points in the sense that

• (i)

$S_{n}^{+}$ acts from left and right on

[TABLE]

by $\alpha(p_{i}):=\sum_{k=1}^{n}u_{ik}\otimes p_{k}$ and $\beta(p_{i}):=\sum_{k=1}^{n}u_{ki}\otimes p_{k}$

• (ii)

and $S_{n}^{+}$ is maximal with these actions, i.e. any other compact matrix quantum group with actions defined as $\alpha$ and $\beta$ is a compact matrix quantum subgroup of $S_{n}^{+}$.

See also [11] for similar questions around quantum symmetries.

3. Quantum automorphism groups of graphs

Wang’s work in the 1990’s was the starting point of the investigations of quantum symmetry phenomena for discrete structures (within Woronowicz’s framework). Note that $n$ points may be viewed as the totally disconnected graph on $n$ vertices. A decade later, Banica and Bichon extended Wang’s approach to a theory of quantum automorphism groups of finite graphs. In the sequel, we restrict to finite graphs having no multiple edges.

3.1. Bichon’s quantum automorphism group of a graph

In 2003, Bichon [5] defined a quantum automorphism group as follows.

Definition 3.1**.**

Let $\Gamma=(V,E)$ be a finite graph with $n$ vertices $V=\{1,...,n\}$ and $m$ edges $E=\{e_{1},...,e_{m}\}$. The quantum automorphism group $G_{aut}^{*}(\Gamma)$ is the compact matrix quantum group $(C(G_{aut}^{*}(\Gamma)),u)$, where $C(G_{aut}^{*}(\Gamma))$ is the universal $C^{*}$-algebra with generators $u_{ij}$, $1\leq i,j\leq n$ and relations

[TABLE]

In the original definition of Bichon, there is actually another relation which is implied by the others:

[TABLE]

Indeed, Relations (3.6) are implied by Relations (3.2), (3.3) and (3.4):

[TABLE]

3.2. Banica’s quantum automorphism group of a graph

Two years later, Banica [1] gave the following definition.

Definition 3.2**.**

Let $\Gamma=(V,E)$ be a finite graph with $n$ vertices and adjacency matrix $\varepsilon\in M_{n}(\{0,1\})$. The quantum automorphism group $G_{aut}^{+}(\Gamma)$ is the compact matrix quantum group $(C(G_{aut}^{+}(\Gamma)),u)$, where $C(G_{aut}^{+}(\Gamma))$ is the universal $C^{*}$-algebra with generators $u_{ij},1\leq i,j\leq n$ and Relations (3.1), (3.2) together with

[TABLE]

which is nothing but $\sum_{k}u_{ik}\varepsilon_{kj}=\sum_{k}\varepsilon_{ik}u_{kj}$.

3.3. Link between the two definitions

It is easy to see ([10, Lemma 3.1.1] or [15, Lemma 6.7]) that Banica’s definition may be expressed as:

[TABLE]

We thus have

[TABLE]

in the sense of compact matrix quantum subgroups, see Section 2.4. Equality holds, if $C(G_{aut}^{*}(\Gamma))$ and $C(G_{aut}^{+}(\Gamma))$ are commutative. Moreover, note that (see Example 2.1):

[TABLE]

Example 3.3**.**

As an example, let $\Gamma$ be the complete graph (i.e. $E=V\times V$). Then:

[TABLE]

For its complement $\Gamma^{c}$ (i.e. $E=\emptyset$), we have:

[TABLE]

3.4. Review of the literature on quantum automorphism groups of graphs

At the moment there are only few articles regarding quantum automorphism groups of graphs. Some results are the following. In [6], Bichon defined the hyperoctahedral quantum group and showed that this group is the quantum automorphism group of some graph. Banica computed the Poincaré series of $G_{aut}^{+}(\Gamma)$ for homogenous graphs with less than eight vertices in [1]. Banica, Bichon and Chenevier considered circulant graphs having $p$ vertices for $p$ prime in [3]. They proved $G_{aut}^{+}(\Gamma)=\mathrm{Aut}(\Gamma)$ if the graph $\Gamma$ does fulfill certain properties. Banica and Bichon investigated $G_{aut}^{+}(\Gamma)$ for vertex-transitive graphs of order less or equal to eleven in [2]. They also computed $G_{aut}^{+}(\Gamma)$ for the direct product, the Cartesian product and the lexicographic product of specific graphs. Chassaniol also studied the lexicographic product of graphs in [7]. In her PhD thesis [10], Fulton studied undirected trees $\Gamma$ such that $\mathrm{Aut}(\Gamma)=\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times...\times\mathbb{Z}_{2}$, where we have $k$ kopies of the cyclic group $\mathbb{Z}_{2}=\mathbb{Z}/2\mathbb{Z}$. She proved $\mathrm{Aut}(\Gamma)=G_{aut}^{*}(\Gamma)=G_{aut}^{+}(\Gamma)$ for $k=1$ and $\mathrm{Aut}(\Gamma)\neq G_{aut}^{*}(\Gamma)=G_{aut}^{+}(\Gamma)$ for $k\geq 2$. See also [4] for links to quantum isometry groups.

3.5. Comparing with the complement of the graph

As can be seen from Section 3.4, the theory of quantum automorphism groups of graphs is still in its infancy. We now provide some basic results on the link between $G_{aut}^{*}(\Gamma)$ and $G_{aut}^{*}(\Gamma^{c})$. Note that while we have

[TABLE]

and

[TABLE]

for all graphs $\Gamma$ (using $\varepsilon_{\Gamma^{c}}=A-\varepsilon_{\Gamma}$ for the adjacency matrices, with $A\in M_{n}(\{1\})$ the matrix filled with units, and $uA=A=Au$ by Relation (3.2)), we may have

[TABLE]

for instance when $\Gamma$ is the complete graph, see Example 3.3.

Lemma 3.4**.**

If $G_{aut}^{*}(\Gamma)\subseteq G_{aut}^{*}(\Gamma^{c})$, then $G_{aut}^{*}(\Gamma)=\mathrm{Aut}(\Gamma)$.

Proof.

Relation (3.5) in $C(G_{aut}^{*}(\Gamma^{c}))$ implies that $u_{ik}$ and $u_{jl}$ commute in$C(G_{aut}^{*}(\Gamma))$ whenever $(i,j)\notin E$ and $(k,l)\notin E$. Together with Relations (3.3), (3.4) and (3.5) in $C(G_{aut}^{*}(\Gamma))$ this yields commutativity of all generators. ∎

Lemma 3.5**.**

If $G_{aut}^{*}(\Gamma^{c})=G_{aut}^{+}(\Gamma^{c})$, then $G_{aut}^{*}(\Gamma)=\mathrm{Aut}(\Gamma)$.

Proof.

We have $G_{aut}^{*}(\Gamma)\subseteq G_{aut}^{+}(\Gamma)=G_{aut}^{+}(\Gamma^{c})=G_{aut}^{*}(\Gamma^{c})$ and apply Lemma 3.4. ∎

The next lemma shows that the quantum automorphism groups of a graph without loops does not change if we add those.

Lemma 3.6**.**

Let $\Gamma=(V,E)$ be a finite graph without loops. Consider $\Gamma^{\prime}=(V,E^{\prime})$ with $E^{\prime}=E\cup\{(i,i),i\in V\}$. It holds

• (i)

$G_{aut}^{+}(\Gamma)=G_{aut}^{+}(\Gamma^{\prime})$,

• (ii)

$G_{aut}^{*}(\Gamma)=G_{aut}^{*}(\Gamma^{\prime})$.

Proof.

For (i), we use $\varepsilon_{\Gamma^{\prime}}=1+\varepsilon_{\Gamma}$, where $1$ is the identity matrix in $M_{n}(\{0,1\})$. Thus, $u\varepsilon_{\Gamma}=\varepsilon_{\Gamma}u$ is equivalent to $u\varepsilon_{\Gamma^{\prime}}=\varepsilon_{\Gamma^{\prime}}u$.

For (ii), all we need to check is that $u_{is(e_{j})}u_{ir(e_{j})}=u_{ir(e_{j})}u_{is(e_{j})}$ is fulfilled in $C(G_{aut}^{*}(\Gamma))$ for all $i\in V$, $e_{j}\in E$, which is true due to Relation (3.1). ∎

3.6. Quantum automorphism groups on four vertices

For a small number of vertices of undirected graphs, a complete classification of $G_{aut}^{*}(\Gamma)$ and $G_{aut}^{+}(\Gamma)$ is possible. For $n\in\{1,2,3\}$, we have $C(S_{n}^{+})=C(S_{n})$, hence $\operatorname{Aut}(\Gamma)=G_{aut}^{*}(\Gamma)=G_{aut}^{+}(\Gamma)$. For $n=4$, we now provide a complete table for graphs having no loops. We restrict to undirected graphs in order to keep it simple. We need the following lemma to compute the quantum automorphism groups.

Lemma 3.7**.**

Let $\Gamma=(V,E)$ be a finite graph with $V=\{1,...,n\}$ and let $e_{j}\in E$. Let $q\in V$ with $s^{-1}(q)=\emptyset$. For the generators of $C(G_{aut}^{+}(\Gamma))$ it holds

[TABLE]

Proof.

By Relations (3.2) and (3.4), we get

[TABLE]

because $(q,i)\notin E$ for all $i\in V$. Likewise, we get $u_{s(e_{j})q}=0$. ∎

In the following, $D_{4}$ denotes the dihedral group defined as

[TABLE]

$H_{2}^{+}$ denotes the hyperoctahedral quantum group defined by Bichon in [6] and $\mathbb{Z}_{2}$ denotes the cyclic group $\mathbb{Z}/2\mathbb{Z}$. The quantum group $\widehat{\mathbb{Z}_{2}*\mathbb{Z}_{2}}=(C^{*}(\mathbb{Z}_{2}*\mathbb{Z}_{2}),u)$ is understood as the compact matrix quantum group with matrix

[TABLE]

where $C^{*}(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ is seen as the universal unital $C^{*}$-algebra generated by two projections $p$ and $q$. Recall that $\operatorname{Aut}(\Gamma)=\operatorname{Aut}(\Gamma^{c})$ and $G_{aut}^{+}(\Gamma)=G_{aut}^{+}(\Gamma^{c})$, where $\Gamma^{c}$ is the complement of $\Gamma$ within the category of graphs having no loops. Parts of the following table were also computed in [2] and [6].

Theorem 3.8**.**

Let $\Gamma$ be an undirected graph on four vertices having no loops and no multiple edges. Then:

[TABLE]

Proof.

For every row of the table, we compute $G_{aut}^{+}(\Gamma)$ and we show $G_{aut}^{+}(\Gamma)=G_{aut}^{*}(\Gamma)$. We then obtain $G_{aut}^{*}(\Gamma^{c})$ by using Lemma 3.5. We label the points of the graphs as follows:

\bullet$$\bullet$$\bullet$$\bullet1234

• (1)

Obvious, see Example 3.3.

• (2)

Let $(u_{ij})_{1\leq i,j\leq 4}$ be the generators of $C(G_{aut}^{+}(\Gamma))$. Lemma 3.7 yields

[TABLE]

With Relations (3.2) we deduce

[TABLE]

Thus

[TABLE]

Since $u_{ij}u_{kl}=u_{kl}u_{ij}$ holds for $(i,k),(j,l)\in\{(1,2),(2,1)\}$ in $C(G_{aut}^{+}(\Gamma))$, we get $G_{aut}^{*}(\Gamma)=G_{aut}^{+}(\Gamma)$.

• (3)

Lemma 3.7 yields

[TABLE]

This implies

[TABLE]

thus $G_{aut}^{+}(\Gamma)$ is commutative and hence $G_{aut}^{+}(\Gamma)=G_{aut}^{*}(\Gamma)=\operatorname{Aut}(\Gamma)=\mathbb{Z}_{2}$.

• (4)

Let $\Delta$ and $\Delta^{\prime}$ be the comultiplication maps of $G_{aut}^{+}(\Gamma)$ and $H_{2}^{+}$, respectively. We first show that these two quantum groups coincide as compact quantum groups, i.e. there is a ∗-isomorphism

[TABLE]

such that $\Delta^{\prime}\circ\varphi=(\varphi\otimes\varphi)\circ\Delta$.

Step 1: The map $\varphi$ exists and we have $\Delta^{\prime}\circ\varphi=(\varphi\otimes\varphi)\circ\Delta$.

From $\varepsilon u=u\varepsilon$ we get

[TABLE]

Define $v_{11}:=u_{11}-u_{12}$, $v_{12}:=u_{13}-u_{14}$, $v_{21}:=u_{31}-u_{32}$ and $v_{22}:=u_{33}-u_{34}$. One can compute that $v_{ij}$, $i,j=1,2$ fulfill the relations of $C(H_{2}^{+})$ and with the universal property we get a *-homomorphism $\varphi:C(H_{2}^{+})\to C(G_{aut}^{+}(\Gamma))$. Since $\Delta^{\prime}\circ\varphi=(\varphi\otimes\varphi)\circ\Delta$ also holds, we get that $G_{aut}^{+}(\Gamma)$ is a quantum subgroup of $H_{2}^{+}$.

Step 2: The map $\varphi$ is a ∗-isomorphism.

Let $(v_{ij})_{i,j=1,2}$ be the generators of $C(H_{2}^{+})$. Define

[TABLE]

One can show that the $(u_{ij})_{1\leq i,j\leq 4}$ fulfill the relations of $C(G_{aut}^{+}(\Gamma))$. The universal property now gives us a *-homomorphism $\varphi^{\prime}:C(G_{aut}^{+}(\Gamma))\to C(H_{2}^{+})$ and $\varphi^{\prime}$ is the inverse of $\varphi$ and vice versa.

Step 3: We have $G_{aut}^{+}(\Gamma)=G_{aut}^{*}(\Gamma)$.

We have seen in Step 1, that

[TABLE]

for all $(i,k),(j,l)\in E$. Thus $G_{aut}^{+}(\Gamma)=G_{aut}^{*}(\Gamma)$.

• (5)

We conclude as in (3).

• (6)

Some direct computations using $\varepsilon u=u\varepsilon$ and Relations (3.2) show

[TABLE]

Thus $G_{aut}^{+}(\Gamma)$ is commutative.

∎

4. Proof of the main result

We now prove the main result of this article (see Section 1) for a finite graph $\Gamma$ with vertices $V=\{1,\ldots,n\}$ and edges $E=\{e_{1},\ldots,e_{m}\}$ having no multiple edges.

Remark 4.1**.**

We define the quantum symmetry group $\textnormal{QSym}(C^{*}(\Gamma))$ of $C^{*}(\Gamma)$ to be the maximal compact matrix quantum group $G$ acting on $C^{*}(\Gamma)$ by $\alpha:C^{*}(\Gamma)\to C(G)\otimes C^{*}(\Gamma)$ and $\beta:C^{*}(\Gamma)\to C(G)\otimes C^{*}(\Gamma)$ as defined in the statement of our main theorem. We thus have to show that $G_{aut}^{+}(\Gamma)$ acts on $C^{*}(\Gamma)$ via $\alpha$ and $\beta$ (see Sections 4.1 and 4.2) and that it is maximal with these actions (see Section 4.3).

4.1. Existence of the maps $\alpha$ and $\beta$

In order to prove that

[TABLE]

defines a $*$-homomorphism, all we have to show is that the relations of $C^{*}(\Gamma)$ hold for $p_{i}^{\prime}$ and $s_{e_{j}}^{\prime}$. We may then use the universal property of $C^{*}(\Gamma)$. The proof for the existence of $\beta$ is analogous.

4.1.1. The $p_{i}^{\prime}$ are mutually orthogonal projections

Obviously, $p_{i}^{\prime}=(p_{i}^{\prime})^{*}$ holds. Moreover, using $p_{k}p_{l}=\delta_{kl}p_{k}$ and Relations (3.1), we have

[TABLE]

4.1.2. The $s_{e_{j}}^{\prime}$ are partial isometries with $(s_{e_{j}}^{\prime})^{*}s_{e_{j}}^{\prime}=p_{r(e_{j})}^{\prime}$

Using $s_{e_{l}}^{*}s_{ei}=\delta_{il}p_{r(e_{i})}$ (see Section 2.3) and Relations (3.1), we have

[TABLE]

By Relations (3.3) we have $u_{r(e_{j})j^{\prime}}u_{s(e_{j})i^{\prime}}u_{r(e_{j})j^{\prime}}=0$ for $(i^{\prime},j^{\prime})\notin E$. This yields

[TABLE]

Using Relations (3.2), we obtain $\sum_{i=1}^{n}u_{s(e_{j})i^{\prime}}=1$ and thus

[TABLE]

4.1.3. We have $\sum_{j:\;s(e_{j})=v}s_{e_{j}}^{\prime}(s_{e_{j}}^{\prime})^{*}=p_{v}^{\prime}$ for $s^{-1}(v)\neq\emptyset$

Using Relations (3.1), we get for $v\in V$ with $s^{-1}(v)\neq\emptyset$:

[TABLE]

Now,

[TABLE]

and for $q\in V$ with $(v,q)\notin E$ we have $u_{vs(e_{l})}u_{qr(e_{l})}=0$ by Relations (3.4). Thus, for any $l\in\{1,\ldots,m\}$, we have using Relations (3.2)

[TABLE]

and hence:

[TABLE]

Since $\Gamma$ has no multiple edges by assumption, $r(e_{i})=r(e_{l})$ and $s(e_{i})=s(e_{l})$ implies $e_{i}=e_{l}$. We thus infer using Relations (3.1):

[TABLE]

Now, for $V^{\prime}:=\{q\in V\;|\;s^{-1}(q)\neq\emptyset\}$, we have, using the relations in $C^{*}(\Gamma)$:

[TABLE]

Since we know that $u_{vq}=0$ for $q\notin V^{\prime}$ by Lemma 3.7, we finally conclude

[TABLE]

This settles the existence of $\alpha$.

4.2. The map $\alpha$ is a left action and $\beta$ is a right action

We only prove this claim for $\alpha$, the proof for $\beta$ being analogous.

4.2.1. $(\Delta\otimes\mathrm{id})\circ\alpha=(\mathrm{id}\otimes\alpha)\circ\alpha$ holds and $\alpha$ is unital

Using Relations (3.3), this is straightforward to check.

It remains to show that

[TABLE]

is dense in $C(G_{aut}^{+}(\Gamma))\otimes C^{*}(\Gamma)$, which we will do in the sequel.

4.2.2. The elements $1\otimes p_{l},1\otimes s_{e_{l}}$ and $1\otimes s_{e_{l}}^{*}$ are in $\mathcal{S}$

Using Relations (3.1) and (3.2) we infer:

[TABLE]

Moreover, for $e_{l}\in E$ we get, using Relations (3.1) and $V^{\prime}:=\{v\in V\;|\;s^{-1}(v)\neq\emptyset\}$:

[TABLE]

We proceed similar to Step 4.1.3. By Relations (3.4), we know $u_{qr(e_{l})}u_{vs(e_{l})}=0$ for $(v,q)\notin E$. Thus, by Relations (3.1) and (3.2) and using that $\Gamma$ has no multiple edges, we obtain:

[TABLE]

Finally, Lemma 3.7 yields $u_{vs(e_{l})}=0$ for $v\notin V^{\prime}$. Hence, using Relations (3.2):

[TABLE]

Define $V^{\prime\prime}:=\{v\in V\;|\;r^{-1}(v)\neq\emptyset\}$. Similar to the computations above, we get

[TABLE]

4.2.3. If $1\otimes x,1\otimes y\in\mathcal{S}$, then also $1\otimes xy\in\mathcal{S}$

The remainder of the proof of Step 4.2 consists in general facts for actions of compact matrix quantum groups.

We may write $1\otimes x\in\mathcal{S}$ and $1\otimes y\in\mathcal{S}$ as

[TABLE]

for some $z_{i},t_{j}\in C^{*}(\Gamma)$ and $w_{i},v_{j}\in C(G_{aut}^{+}(\Gamma))$. Therefore

[TABLE]

4.2.4. $\mathcal{S}$ is dense in $C(G_{aut}^{+}(\Gamma))\otimes C^{*}(\Gamma)$

Summarizing, we get that $1\otimes w\in\mathcal{S}$ for all monomials $w$ in $p_{i},s_{e_{j}},s_{e_{j}}^{*}$, $1\leq i\leq n$, $1\leq j\leq m$. Since $\alpha$ is unital, we also have:

[TABLE]

We conclude that $\mathcal{S}$ is dense in $C(G_{aut}^{+}(\Gamma))\otimes C^{*}(\Gamma)$, which settles Step 4.2.

4.3. The quantum group $G_{aut}^{+}(\Gamma)$ acts maximally on $C^{*}(\Gamma)$

For proving the maximality, let $G=(C(G),u)$ be another compact matrix quantum group acting on $C^{*}(\Gamma)$ by $\alpha^{\prime}:C^{*}(\Gamma)\to C(G)\otimes C^{*}(\Gamma)$ and $\beta^{\prime}:C^{*}(\Gamma)\to C(G)\otimes C^{*}(\Gamma)$ in the way $G_{aut}^{+}(\Gamma)$ acts on $C^{*}(\Gamma)$ via $\alpha$ and $\beta$. We want to show that there is a *-homomorphism $C(G_{aut}^{+}(\Gamma))\to C(G)$ sending generators to generators. Thus, we need to compute that the generators $u_{ij}$ of $C(G)$ fulfill the relations of $C(G_{aut}^{+}(\Gamma))$.

4.3.1. The Relations (3.1) hold in $C(G)$

The equation

[TABLE]

yields $u_{ij}=u_{ij}^{*}$ after multiplying from the left with $1\otimes p_{j}$. We also have

[TABLE]

from which we infer $u_{ji}u_{ki}=\delta_{jk}u_{ji}$. Using $\beta^{\prime}$, we also obtain $u_{ij}u_{ik}=\delta_{jk}u_{ij}$.

4.3.2. The Relations (3.2) hold in $C(G)$

From

[TABLE]

we deduce $\sum_{i=1}^{n}u_{ik}=1$, and likewise $\sum_{i=1}^{n}u_{ki}=1$ using $\beta^{\prime}$.

4.3.3. The Relations (3.3) hold in $C(G)$

Using $s_{e_{l}}^{*}s_{e_{t}}=\delta_{lt}p_{r(e_{l})}$ (see Section 2.3) and Relations (3.1) in $C(G)$, we obtain for any $j$:

[TABLE]

Multiplication with $1\otimes p_{k}$ yields:

[TABLE]

If $r^{-1}(k)=\emptyset$, then $u_{r(e_{j})k}=0$ and hence $u_{s(e_{j})i}u_{r(e_{j})k}=u_{r(e_{j})k}u_{s(e_{j})i}=0$ for all $i\in V$.

Otherwise, if $r^{-1}(k)\neq\emptyset$, we use Relations (3.1) and (3.2) in $C(G)$ and get

[TABLE]

and therefore

[TABLE]

Since

[TABLE]

holds, the above is a vanishing sum of positive elements – and hence each summand vanishes. This yields $u_{s(e_{j})i}u_{r(e_{j})k}=0$ for all $(i,k)\notin E$.

4.3.4. The Relations (3.4) hold in $C(G)$

The argument is analogous to the one for proving Relations (3.3) when replacing $\alpha^{\prime}$ by $\beta^{\prime}$.

The proof of the main theorem is complete.

Remark 4.2**.**

Let $\Gamma$ be a finite graph with $n$ vertices $V=\{1,...,n\}$ and $m$ edges $E=\{e_{1},...,e_{m}\}$. In [5], Bichon showed that $G_{aut}^{*}(\Gamma)$ is the quantum symmetry group of $\Gamma$ in his sense, where

[TABLE]

define actions of $G_{aut}^{*}(\Gamma)$ on $C(V)$ and $C(E)$, respectively. Those actions inspired us, how an action of a compact matrix quantum group on $C^{*}(\Gamma)$ should look like. However, note that edges in the commutative $C^{*}$-algebra $C(E)$ of continuous functions on $E$ are represented as projections unlike in the case of $C^{*}(\Gamma)$. Therefore, the quantum symmetry group of $C^{*}(\Gamma)$ is $G_{aut}^{+}(\Gamma)$ rather than $G_{aut}^{*}(\Gamma)$. On the other hand, if we consider the quotient of $C^{*}(\Gamma)$ by the relations $s_{e}=s_{e}^{*}$, its quantum symmetry group is $G_{aut}^{*}(\Gamma)$. Indeed, selfadjointness of $s_{e}$ yields

[TABLE]

from which we obtain Relations (3.5) by multiplication with $(1\otimes s_{e_{i}}^{*})$ from the left.