QUANTUM AUTOMORPHISMS OF FOLDED CUBE GRAPHS

(1)

ANNALES DE

L’INSTITUT FOURIER

LesAnnales de l’institut Fouriersont membres du Centre Mersenne pour l’édition scientifique ouverte

Simon Schmidt

Quantum automorphisms of folded cube graphs Tome 70, n^o3 (2020), p. 949-970.

<http://aif.centre-mersenne.org/item/AIF_2020__70_3_949_0>

Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France.

http://creativecommons.org/licenses/by-nd/3.0/fr/

(2)

QUANTUM AUTOMORPHISMS OF FOLDED CUBE GRAPHS

by Simon SCHMIDT (*)

Abstract. —We show that the quantum automorphism group of the Clebsch graph isSO⁻¹₅ . This answers a question by Banica, Bichon and Collins from 2007.

More general for odd n, the quantum automorphism group of the foldedn-cube graph isSOn⁻¹. Furthermore, we show that if the automorphism group of a graph contains a pair of disjoint automorphisms this graph has quantum symmetry.

Résumé. —On démontre que le groupe quantique d’automorphismes du graphe de Clebsch estSO⁻¹₅ ce qui répond à une question de Banica, Bichon et Collins de 2007. En général, pour des valeurs impaires de n, le groupe quantique d’automor- phisme du graphe du n-cube plié estSOn⁻¹. En plus, on démontre qu’un graphe possède des symétries quantiques, si son groupe d’automorphismes contient une paire d’automorphismes disjoints.

Introduction

The concept of quantum automorphism groups of finite graphs was introduced by Banica and Bichon in [1, 9]. It generalizes the classical automorphism groups of graphs within the framework of compact matrix quantum groups. We say that a graph has no quantum symmetry if the quantum automorphism group coincides with its usual automorphism group. A nat- ural question is: When does a graph have no quantum symmetry? This has been studied in [5] for some graphs on pvertices, pprime, and more recently the author showed in [16] that the Petersen graph does not have quantum symmetry. Also Lupini, Mančinska and Roberson proved that almost all graphs have trivial quantum automorphim group in [12], which implies that almost all graphs do not have quantum symmetry.

Keywords:finite graphs, graph automorphisms, automorphism groups, quantum automorphisms, quantum groups, quantum symmetries.

2020Mathematics Subject Classification:46LXX, 20B25, 05CXX.

(*) The author is supported by the DFG projectQuantenautomorphismen von Graphen.

(3)

In this article we develop a tool for detecting quantum symmetries namely we show that a graph has quantum symmetry if its automorphism group contains a pair of disjoint automorphisms (Theorem 2.2). As an example, we apply it to the Clebsch graph and obtain that it does have quantum symmetry (Corollary 3.2).

We even go further and prove that the quantum automorphism group of the Clebsch graph isSO₅⁻¹, theq-deformation atq=−1 ofSO5, answering a question from [7]. For this we use the fact that the Clebsch graph is the folded 5-cube graph. This can be pushed further to more general folded n-cube graphs: In [6], two generalizations of the hyperoctahedral group Hn are given, one of them being O_n⁻¹ as quantum symmetry group of the hypercube graph. To prove thatO⁻¹_n is the quantum symmetry group of the hypercube graph, Banica, Bichon and Collins used the fact that the hypercube graph is a Cayley graph. It is also well known that the folded cube graph is a Cayley graph. We use similar techniques as in [6] to show that for oddn, the quantum symmetry group of the foldedn-cube graph is SO⁻¹_n which is the quotient ofO⁻¹_n by some quantum determinant condition (Theorem 5.9). This constitutes our main result.

Acknowledgments

The author is grateful to his supervisor Moritz Weber for proofreading the article and for numerous discussions on the topic. He also wants to thank Julien Bichon and David Roberson for helpful comments and sug- gestions. Furthermore, he thanks the referee for useful comments and for correcting a mistake in the proof of Theorem 2.2.

1. Preliminaries

1.1. Compact matrix quantum groups

We start with the definition of compact matrix quantum groups which were defined by Woronowicz [21, 22] in 1987. See [13, 18] for recent books on compact quantum groups.

Definition 1.1. — A compact matrix quantum group G is a pair (C(G), u), whereC(G)is a unital (not necessarily commutative)C^∗-algebra

(4)

which is generated by u_ij, 1 6 i, j 6 n, the entries of a matrix u ∈ Mn(C(G)). Moreover, the *-homomorphism ∆ : C(G) → C(G)⊗C(G), u_ij 7→ Pn

k=1u_ik⊗u_kj must exist, and u and its transpose u^t must be invertible.

An important example of a compact matrix quantum group is the quantum symmetric groupS_n⁺due to Wang [20]. It is the compact matrix quantum group, where

C(S_n⁺) :=C^∗ uij,16i, j6n

uij =u^∗_ij =u²_ij,X

l

uil=X

l

uli= 1

! .

An action of a compact matrix quantum group on aC^∗-algebra is defined as follows [14, 20].

Definition 1.2. — LetG= (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

*-homomorphismα:B→B⊗C(G)such that (1) (id⊗∆)◦α= (α⊗id)◦α

(2) α(B)(1⊗C(G))is linearly dense inB⊗C(G).

In [20], Wang showed thatS_n⁺ is the universal compact matrix quantum group acting onXn={1, . . . , n}. This action is of the formα:C(Xn)→ C(Xn)⊗C(S_n⁺),

α(ei) =X

j

ej⊗uji.

1.2. Quantum automorphism groups of finite graphs In 2005, Banica [1] gave the following definition of a quantum automorphism group of a finite graph.

Definition 1.3. — Let Γ = (V, E) be a finite graph with n vertices and adjacency matrixε∈Mn({0,1}). The quantum automorphism group G⁺_aut(Γ) is the compact matrix quantum group (C(G⁺_aut(Γ)), u), where C(G⁺_aut(Γ)) is the universal C^∗-algebra with generators uij,1 6 i, j 6 n and relations

uij =u^∗_ij =u²_ij 16i, j6n, (1.1)

n

X

l=1

u_il= 1 =

n

X

l=1

u_li, 16i6n,

(1.2)

uε=εu, (1.3)

where(1.3)is nothing butP

kuikεkj=P

kεikukj.

(5)

Remark 1.4. — There is another definition of a quantum automorphism group of a finite graph by Bichon in [9], which is a quantum subgroup of G⁺_aut(Γ). But this article concerns G⁺_aut(Γ). See [17] for more on quantum automorphism groups of graphs.

The next definition is due to Banica and Bichon [3]. We denote by Gaut(Γ) the usual automorphism group of a graph Γ.

Definition 1.5. — LetΓ = (V, E)be a finite graph. We say thatΓhas no quantum symmetryifC(G⁺_aut(Γ))is commutative, or equivalently

C(G⁺_aut(Γ)) =C(Gaut(Γ)).

IfC(G⁺_aut(Γ))is non-commutative, we say thatΓ hasquantum symmetry.

Note thatG_aut(Γ)⊆G⁺_aut(Γ), so in general a graph Γ has more quantum symmetries than symmetries.

1.3. Compact matrix quantum groups acting on graphs

An action of a compact matrix quantum group on a graph is an action on the functions on the vertices, but with additional structure. This concept was introduced by Banica and Bichon [1, 9].

Definition 1.6. — LetΓ = (V, E)be a finite graph andGbe a compact matrix quantum group. Anaction ofGon Γis an action ofGonC(V)such that the magic unitary matrix(vij)₁₆_i,j₆_|V_|associated to the formular

α(e_i) =

|V|

X

j=1

e_j⊗v_ji

commutes with the adjacency matrix, i.evε=εv.

Remark 1.7. — If G acts on a graph Γ, then we have a surjective

*-homomorphismϕ:C(G⁺_aut(Γ))→C(G),u7→v.

The following theorem shows that commutation with the magic unitary uyields invariant subspaces.

Theorem 1.8 ([1, Theorem 2.3]). — Letα:C(X_n)→C(X_n)⊗C(G), α(ei) =P

jej⊗vji be an action, whereGis a compact matrix quantum group and letKbe a linear subspace ofC(X_n). The matrix(v_ij)commutes with the projection ontoKif and only if α(K)⊆K⊗C(G).

(6)

Looking at the spectral decomposition of the adjacency matrix, we see that this action preserves the eigenspaces of the adjacency matrix.

Corollary 1.9. — LetΓ = (V, E)be an undirected finite graph with adjacency matrixε. The actionα:C(V)→C(V)⊗C(G⁺_aut(Γ)), α(ei) = P

je_j⊗u_ji, preserves the eigenspaces ofε, i.e. α(E_λ)⊆E_λ⊗C(G⁺_aut(Γ)) for all eigenspacesEλ.

Proof. — It follows from the spectral decomposition that every projec- tionPE_λ ontoEλ is a polynomial inε. Thus it commutes with the fundamental corepresentationuand Theorem 1.8 yields the assertion.

1.4. Fourier transform

One can obtain a C^∗-algebra from the group Zⁿ2 by either considering the continuous functions C(Zⁿ2) over the group or the group C^∗-algebra C^∗(Zⁿ2). SinceZⁿ2 is abelian, we know thatC(Zⁿ2)∼=C^∗(Zⁿ2) by Pontryagin duality. This isomorphism is given by the Fourier transform and its inverse.

We write

Zⁿ2 ={tⁱ₁¹. . . tⁱ_nⁿ|i1, . . . , in∈ {0,1}}, C(Zⁿ2) = span(e_ti1

1 ...tⁱⁿ_n |tⁱ₁¹. . . tⁱ_nⁿ∈Zⁿ2),

C^∗(Zⁿ2) =C^∗(t1, . . . , tn|ti=t^∗_i, t²_i = 1, titj =tjti), where

e_ti1

1...tⁱⁿ_n :Zⁿ2 →C, e_ti1

1 ...tⁱⁿ_n (t^j₁¹. . . t^j_nⁿ) =δi₁j₁. . . δi_nj_n. The proof of the following proposition can be found in [2] for example.

Proposition 1.10. — The *-homomorphisms ϕ:C(Zⁿ2)→C^∗(Zⁿ2), e_ti1

1 ...tⁱⁿ_n → 1 2ⁿ

1

X

j₁,...,j_n=0

(−1)ⁱ¹^j¹^+···+iⁿ^jⁿt^j₁¹. . . t^j_nⁿ

and

ψ:C^∗(Zⁿ2)→C(Zⁿ2), tⁱ₁¹. . . tⁱ_nⁿ→

1

X

j1,...,jn=0

(−1)ⁱ¹^j¹^+···+iⁿ^jⁿe_tj1 1 ...t^jn_n , where i₁, . . . , i_n ∈ {0,1}, are inverse to each other. The map ϕ is called Fourier transform, the mapψis called inverse Fourier transform.

(7)

2. A criterion for a graph to have quantum symmetry In this section, we show that a graph has quantum symmetry if the automorphism group of the graph contains a certain pair of permutations.

For this we need the following definition.

Definition 2.1. — LetV ={1, . . . , r}. We say that two permutations σ:V →V andτ:V →V aredisjoint, ifσ(i)6=iimpliesτ(i) =iand vice versa, for alli∈V.

Theorem 2.2. — Let Γ = (V, E) be a finite graph without multiple edges. If there exist two non-trivial, disjoint automorphismsσ, τ ∈Gaut(Γ), ord(σ) = n,ord(τ) = m, then we get a surjective *-homomorphism ϕ : C(G⁺_aut(Γ)) → C^∗(Zn∗Zm). In particular, Γ has quantum symmetry if n, m>2.

Proof. — Letσ, τ ∈G_aut(Γ) be non-trivial disjoint automorphisms with ord(σ) =n,ord(τ) =m. Define

A:=C^∗ p₁, . . . , p_n, q₁, . . . , q_m

pk=p^∗_k=p²_k, ql=q^∗_l =q²_l,

n

X

k=1

p_k= 1 =

m

X

l=1

q_l

!

∼=C^∗(Zn∗Zm).

We want to use the universal property to get a surjective *-homomorphism ϕ:C(G⁺_aut(Γ))→A. This yields the non-commutativity of G⁺_aut(Γ), since pk, qldo not have to commute. In order to do so, define

u⁰:=

m

X

l=1

τ^l⊗q_l+

n

X

k=1

σ^k⊗p_k−id_M_r₍_C_)⊗A∈M_r(C)⊗A∼= Mr(A), where τ^l, σ^k denote the permutation matrices corresponding to τ^l, σ^k ∈ G_aut(Γ). This yields

u⁰_ij=

m

X

l=1

δ_jτl(i)⊗ql+

n

X

k=1

δ_jσk(i)⊗pk−δij ∈C⊗A∼=A.

Now, we show that u⁰ does fulfill the relations of u ∈ Mr(C)⊗A, the fundamental representation ofG⁺_aut(Γ). Since we haveτ^l, σ^k ∈G_aut(Γ), it holds τ^lε = ετ^l and σ^kε = εσ^k for all 1 6 l 6 m, 1 6 k 6 n, where ε denotes the adjacency matrix of Γ. Therefore, we have

u⁰(ε⊗1) =

m

X

l=1

τ^l⊗ql+

n

X

k=1

σ^k⊗pk−idM_r(C)⊗A

! (ε⊗1)

=

m

X

l=1

τ^lε⊗ql+

n

X

k=1

σ^kε⊗pk−(ε⊗1)

(8)

=

m

X

l=1

ετ^l⊗q_l+

n

X

k=1

εσ^k⊗p_k−(ε⊗1)

= (ε⊗1)

m

X

l=1

τ^l⊗ql+

n

X

k=1

σ^k⊗pk−id_M_r₍_C_)⊗A

!

= (ε⊗1)u⁰. Furthermore, it holds

r

X

i=1

u⁰_ji=

r

X

i=1 m

X

l=1

δ_iτl(j)⊗ql+

n

X

k=1

δ_iσk(j)⊗pk

!

−1⊗1

= 1⊗

m

X

l=1

ql

! + 1⊗

n

X

k=1

pk

!

−1⊗1

= 1⊗1.

A similar computation showsPr

i=1u⁰_ij = 1⊗1. Sinceτ andσare disjoint, we have

u⁰_ij=

m

X

l=1

δ_jτl(i)⊗ql+

n

X

k=1

δ_jσk(i)⊗pk−δij =









 P

k∈Nijpk, ifσ(i)6=i P

l∈Mijql, ifτ(i)6=i δ_ij, otherwise, whereNij={k∈ {1, . . . , n};σ^k(i) =j}, Mij ={l∈ {1, . . . , m};τ^l(i) =j}.

Thus, all entries ofu⁰ are projections. By the universal property, we get a

*-homomorphismϕ:C(G⁺_aut(Γ))→A, u7→u⁰.

It remains to show that ϕ is surjective. We know ord(σ) = n. By de- composing σ into a product of disjoint cycles, we see that there exist s1, . . . , sa ∈V such that for allk16=k2, k1, k2∈ {1, . . . , n}, we have

(σ^k¹(s₁), . . . , σ^k¹(s_a))6= (σ^k²(s₁), . . . , σ^k²(s_a)).

By similar considerations, there existt1, . . . , tb∈V such that (τ^l¹(t₁), . . . , τ^l¹(t_b))6= (τ^l²(t₁), . . . , τ^l²(t_b)) forl16=l2, l1, l2∈ {1, . . . , m}. Therefore, we have

ϕ(u_s₁_σk(s₁). . . u_s_a_σk(s_a)=u⁰_s

1σ^k(s1). . . u⁰_s

aσ^k(sa)=pk, ϕ(u_t₁_τl(t1). . . u_t_b_τl(tb)) =u⁰_t

1τ^l(t₁). . . u⁰_t

bτ^l(t_b) =q_l

for allk∈ {1, . . . , n}, l∈ {1, . . . , m} and sinceAis generated byp_k andq_l,

ϕis surjective.

(9)

Remark 2.3. — Let K₄ be the full graph on 4 points. We know that Gaut(K4) = S4 and G⁺_aut(K4) = S₄⁺. We have disjoint automorphisms in S₄, where for example σ = (12), τ = (34) ∈ S₄ give us the well known surjective *-homomorphism

ϕ:C(S₄⁺)→C^∗(p, q|p=p^∗=p², q=q^∗=q²),

u7→







p 1−p 0 0

1−p p 0 0

0 0 q 1−q

0 0 1−q q





 ,

yielding the non-commutativity ofS₄⁺.

Remark 2.4. — Let Γ = (V, E) be a finite graph without multiple edges, where there exist two non-trivial, disjoint automorphismsσ, τ ∈Gaut(Γ).

To show that Γ has quantum symmetry it is enough to see that we have the surjective *-homomorphism

ϕ⁰:C(G⁺_aut(Γ))→C^∗(p, q|p=p^∗=p², q=q^∗=q²), u7→σ⊗p+τ⊗q+ idM_r(C)⊗(1−q−p).

Remark 2.5. — At the moment, we do not have an example of a graph Γ, whereGaut(Γ) does not contain two disjoint automorphisms but the graph has quantum symmetry.

3. The Clebsch graph has quantum symmetry

As an application of Theorem 2.2, we show that the Clebsch graph does have quantum symmetry. In Section 5, we will study the quantum automorphism group of this graph:

Figure 3.1. The Clebsch graph

(10)

Proposition 3.1. — The Clebsch graph has disjoint automorphisms.

Proof. — We label the graph as follows

1

2

3

4 5

6

7 8 9

10 11

12

13

14 15

16

Then we get two non-trivial disjoint automorphisms of this graph σ= (2 3)(6 7)(10 11)(14 15),

τ= (1 4)(5 8)(9 12)(13 16).

Corollary 3.2. — The Clebsch graph does have quantum symmetries, i.e.C(G⁺_aut(ΓClebsch))is non-commutative.

Proof. — By Theorem 2.2, we get thatC(G⁺_aut(ΓClebsch)) is non-commutative. Looking at the proof of Theorem 2.2, we get the surjective *- homomorphism ϕ : C(G⁺_aut(ΓClebsch)) → C^∗(p, q|p = p^∗ = p², q = q^∗

=q²),

u7→u⁰=







u⁰⁰ 0 0 0

0 u⁰⁰ 0 0 0 0 u⁰⁰ 0 0 0 0 u⁰⁰





 ,

where

u⁰⁰=







q 0 0 1−q

0 p 1−p 0

0 1−p p 0

1−q 0 0 q







.

Remark 3.3.

(1) The Clebsch graph is the folded 5-cube graph, which will be introduced in Section 5. There we will study the quantum automorphism group for (2m+1)-folded cube graphs going far beyond Corol- lary 3.2.

(11)

(2) Using Theorem 2.2, it is also easy to see that the folded cube graphs have quantum symmetry, but this will also follow from our main result (Theorem 5.9).

4. The quantum group SO_n⁻¹

Now, we will have a closer look at the quantum group SO_n⁻¹, but first we defineO⁻¹_n , which appeared in [6] as the quantum automorphism group of the hypercube graph. For both it is immediate to check that the comul- tiplication ∆ is a *-homomorphism.

Definition 4.1. — We defineO⁻¹_n to be the compact matrix quantum group(C(O⁻¹_n ), u), whereC(O⁻¹_n )is the universal C^∗-algebra with gener- atorsuij,16i, j6nand relations

uij=u^∗_ij, 16i, j6n,

(4.1)

n

X

k=1

u_iku_jk =

n

X

k=1

u_kiu_kj=δ_ij, 16i, j6n, (4.2)

uijuik=−uikuij, ujiuki=−ukiuji, k6=j, (4.3)

uijukl=ukluij, i6=k, j6=l.

(4.4)

For n = 3, SO⁻¹_n appeared in [4], where Banica and Bichon showed SO⁻¹₃ =S₄⁺. Our main result in this paper is that fornodd,SO_n⁻¹ is the quantum automorphism group of the foldedn-cube graph.

Definition 4.2. — We defineSO_n⁻¹to be the compact matrix quantum group(C(SO⁻¹_n ), u), whereC(SO_n⁻¹)is the universalC^∗-algebra with gen- eratorsuij,16i, j6n, Relations(4.1)–(4.4)and

X

σ∈S_n

u_σ(1)1. . . u_σ(n)n = 1.

(4.5)

Lemma 4.3. — Let(u_ij)₁₆_i,j₆_n be the generators ofC(SO_n⁻¹). Then X

σ∈Sn

uσ(1)1. . . u_{σ(n−1)n−1}uσ(n)k= 0 fork6=n.

Proof. — Let 16k6n−1. Using Relations (4.3) and (4.4) we get u_σ(1)1. . . u_σ(k)k. . . u_{σ(n−1)n−1}u_σ(n)k

=−u_σ(1)1. . . u_σ(n)k. . . u_{σ(n−1)n−1}u_σ(k)k

=−u_τ(1)1. . . u_τ(k)k. . . u_{τ(n−1)n−1}u_τ(n)k

(12)

forτ =σ◦(k n)∈S_n. Therefore the summands corresponding toσand τ

sum up to zero. The result is then clear.

The next lemma gives an equivalent formulation of Relation (4.5). One direction is a special case of [19, Lemma 4.6].

Lemma 4.4. — LetAbe aC^∗-algebra and letuij ∈Abe elements that fulfill Relations(4.1)–(4.4). Letj ∈ {1, . . . , n}and define

Ij={(i1, . . . , i_n−1)∈ {1, . . . , n}ⁿ⁻¹|ia 6=ib fora6=b, is6=j for alls}.

The following are equivalent (1) We have

1 = X

σ∈Sn

uσ(1)1. . . uσ(n)n.

(2) It holds

ujn= X

(i₁,...,in−1)∈Ij

ui11. . . uin−1n−1, 16j6n.

Proof. — We first show that (2) implies (1). It holds

1 =

n

X

j=1

u²_jn=

n

X

j=1

X

(i₁,...,in−1)∈Ij

u_i₁₁. . . u_i_n−1_n−1u_jn,

where we used Relation (4.2) and (2). Furthermore, we have

n

X

j=1

X

(i₁,...,i_n−1)∈Ij

ui₁1. . . uin−1n−1ujn= X

i1,...,in; ia6=ibfora6=b

ui₁1. . . ui_nn

= X

σ∈S_n

u_σ(1)1. . . u_σ(n)n

and thus (2) implies (1).

Now we show that (1) implies (2). We have

ujn= X

σ∈Sn

u_σ(1)1. . . u_σ(n)nujn=

n

X

k=1

X

σ∈Sn

u_σ(1)1. . . u_{σ(n−1)n−1}u_σ(n)kujk,

(13)

sinceP

σ∈S_nu_σ(1)1. . . u_{σ(n−1)n−1}u_σ(n)ku_jk = 0 for k 6=n by Lemma 4.3.

We get

n

X

k=1

X

σ∈Sn

u_σ(1)1. . . u_{σ(n−1)n−1}u_σ(n)kujk

= X

σ∈Sn

u_σ(1)1. . . u_{σ(n−1)n−1}

n

X

k=1

u_σ(n)kujk

= X

σ∈Sn

uσ(1)1. . . u_{σ(n−1)n−1}δσ(n)j

= X

(i₁,...,in−1)∈Ij

ui₁1. . . uin−1n−1,

where we used Relation (4.2) and we obtain

u_jn= X

(i₁,...,in−1)∈I_j

u_i₁₁. . . u_i_n−1_n−1.

We now discuss representations of SO_2m+1⁻¹ . For definitions and back- ground for this proposition, we refer to [8, 10, 15].

Proposition 4.5. — The category of corepresentations of SO_2m+1⁻¹ is tensor equivalent to the category of representations ofSO2m+1.

Proof. — We first show thatC(SO⁻¹_2m+1) is a cocycle twist ofC(SO2m+1) by proceeding like in [10, Section 4]. Take the unique bicharacterσ:Z^2m2 × Z^2m2 → {±1}with

σ(t_i, t_j) =−1 =−σ(tj, t_i), for 16i < j 62m, σ(ti, ti) = (−1)^m, for 16i62m+ 1, σ(ti, t2m+1) = (−1)^m−i=−σ(t2m+1, ti), for 16i62m, where we use the identification

Z^2m2 =ht1, . . . , t2m+1|t²_i = 1, titj =tjti, t2m+1=t1. . . t2mi.

LetH be the subgroup of diagonal matrices inSO_2m+1having±1 entries.

We get a surjective *-homomorphism

π:C(SO_2m+1)→C^∗(Z^2m2 ) uij 7→δijti

by restricting the functions onSO_2m+1toH and using Fourier transform.

Thus we can form the twisted algebra C(SO2m+1)^σ, where we have the

(14)

multiplication

[uij][ukl] =σ(ti, tk)σ⁻¹(tj, tl)[uijukl] =σ(ti, tk)σ(tj, tl)[uijukl].

We see that the generators [uij] of C(SO2m+1)^σ fulfill the same relations as the generators ofC(SO⁻¹_2m+1) and thus we get an surjective *-homomorphism ϕ : C(SO_2m+1⁻¹ ) → C(SO2m+1)^σ, uij 7→ [uij]. This is an isomorphism for example by using Theorem 3.5 of [11]. Now, Corollary 1.4 and

Proposition 2.1 of [8] yield the assertion.

5. Quantum automorphism groups of folded cube graphs In what follows, we will introduce folded cube graphs F Qn and show that for oddn, the quantum automorphism group of F Qn isSO⁻¹_n .

5.1. The folded n-cube graph F Q_n

Definition 5.1. — The foldedn-cube graphF Qnis the graph with ver- tex setV ={(x1, . . . , x_n−1)|x_i∈ {0,1}}, where two vertices(x₁, . . . , x_n−1) and(y1, . . . , y_n−1)are connected if they differ at exactly one position or if (y₁, . . . , y_n−1) = (1−x₁, . . . ,1−x_n−1).

Remark 5.2. — To justify the name, one can obtain the folded n-cube graph by identifing every opposite pair of vertices from then-hypercube graph.

5.2. The folded cube graph as Cayley graph

It is known that the folded cube graphs are Cayley graphs, we recall this fact in the next lemma.

Lemma 5.3. — The folded n-cube graph F Q_n is the Cayley graph of the group Zⁿ⁻¹2 = ht1, . . . tni, where the generators ti fulfill the relations t²_i = 1, titj =tjti, tn=t₁. . . t_n−1.

Proof. — Consider the Cayley graph ofZⁿ⁻¹2 =ht1, . . . t_ni. The vertices are elements ofZⁿ⁻¹2 , which are products of the formg=tⁱ₁¹. . . tⁱ_n−1ⁿ⁻¹. The exponents are in one to one correspondence to (x₁, . . . , x_n−1), x_i ∈ {0,1}, thus the vertices of the Cayley graph are the vertices of the foldedn-cube

(15)

graph. The edges of the Cayley graph are drawn between vertices g, h, where g =hti for somei. Fork ∈ {1, . . . , n−1}, the operation h→htk

changes thek-th exponent to 1−i_k, so we get edges between vertices that differ at exactly one expontent. The operationh→htn takes t^j₁¹. . . t^j_n−1ⁿ⁻¹ tot^1−j₁ ¹. . . t^1−j_n−1ⁿ⁻¹, thus we get the remaining edges of F Q_n.

5.3. Eigenvalues and Eigenvectors of F Qn

We will now discuss the eigenvalues and eigenvectors of the adjacency matrix ofF Q_n.

Lemma 5.4. — The eigenvectors and corresponding eigenvalues ofF Q_n are given by

w_i₁_...i_n−1 =

1

X

j₁,...,jn−1=0

(−1)ⁱ¹^j¹^+···+iⁿ⁻¹^jⁿ⁻¹e

t^j₁¹...t^jn−1_n−1

λi₁...in−1 = (−1)ⁱ¹+· · ·+ (−1)ⁱⁿ⁻¹+ (−1)ⁱ¹^+···+iⁿ⁻¹,

when the vector space spanned by the vertices of F Q_n is identified with C(Zⁿ⁻¹2 ).

Proof. — Let ε be the adjacency matrix of F Qn. Then we know for a vertexpand a functionf on the vertices that

εf(p) = X

q;(q,p)∈E

f(q).

This yields

εet^j₁¹...t^jn−1_n−1 =

n

X

k=1

et_kt^j₁¹...t^jn−1_n−1

=e

t^j₁^{1 +1}...t^jn−1_n−1 +· · ·+e

t^j₁¹...t^{jn−1 +1}_n−1 +e

t^j₁^{1 +1}...t^{jn−1 +1}_n−1 . For the vectors in the statement we get

εwi₁...in−1 = X

j₁,...,j_n−1

(−1)ⁱ¹^j¹^+···+iⁿ⁻¹^jⁿ⁻¹εe

t^j₁¹...t^jn−1_n−1

=

n−1

X

s=1

X

j1,...,jn−1

(−1)ⁱ¹^j¹^+···+iⁿ⁻¹^jⁿ⁻¹e

t^j₁¹...t^js+1_s ...t^jn−1_n−1

+ X

j₁,...,j_n−1

(−1)ⁱ¹^j¹^+···+iⁿ⁻¹^jⁿ⁻¹e

t^j₁^{1 +1}...t^{jn−1 +1}_n−1 .

(16)

Using the index shiftj_s⁰ =j_s+ 1 mod 2, fors∈ {1, . . . n−1}, we get εwi₁...in−1 =

n−1

X

s=1

X

j1,...j⁰_s,...jn−1

(−1)ⁱ¹^j¹^+···+i^s^(j^s⁰^+1)+···+iⁿ⁻¹^jⁿ⁻¹e

t^j₁¹...t^j_s⁰^s...t^jn−1_n−1

+ X

j⁰₁,...,j_n−1⁰

(−1)ⁱ¹^(j¹⁰^+1)+···+iⁿ⁻¹^(jⁿ⁻¹⁰ ⁺¹⁾e

t^j

0 1 1 ...t

j0 n−1 n−1

=

n−1

X

s=1

X

j₁,...,j_s⁰,...,j_n−1

(−1)ⁱ^s(−1)ⁱ¹^j¹^+···+iⁿ⁻¹^jⁿ⁻¹e

t^j₁¹...t^j

0s s...t^jn−1_n−1

+ X

j⁰₁,...,j_n−1⁰

(−1)ⁱ¹^+···+iⁿ⁻¹(−1)ⁱ¹^j⁰¹^+···+iⁿ⁻¹^jⁿ⁻¹⁰ e

t^j

0 1 1 ...t^j

0 n−1 n−1

= ((−1)ⁱ¹+· · ·+ (−1)ⁱⁿ⁻¹+ (−1)ⁱ¹^+···+iⁿ⁻¹)w_i₁_...i_n−1

=λi₁...in−1wi₁...in−1.

Since those are 2ⁿ⁻¹ vectors that are linearly independent, the assertion

follows.

The following lemma shows what the eigenvectors look like if we identify the vector space spanned by the vertices ofF Qn withC^∗(Zⁿ⁻¹2 ).

Lemma 5.5. — In C^∗(Zⁿ⁻¹2 ) = C^∗(t1, . . . , tn|t²_i = 1, titj = tjti, tn = t₁. . . t_n−1)the eigenvectors ofF Q_n are

wpi₁...in−1 =tⁱ₁¹. . . tⁱ_n−1ⁿ⁻¹

corresponding to the eigenvaluesλ_i₁_...i_n−1 from Lemma 5.4.

Proof. — We obtain wpi₁...i_n−1 by using the Fourier transform (see Sec-

tion 1.4) onwi1...in−1 from Lemma 5.4.

Note that certain eigenvalues in Lemma 5.4 coincide. We get a better description of the eigenvalues and eigenspaces ofF Qn in the next lemma.

Lemma 5.6. — The eigenvalues of F Qn are given by λk =n−2k for k∈2Z∩ {0, . . . , n}. The eigenvectorstⁱ₁¹. . . tⁱ_n−1ⁿ⁻¹ corresponding toλ_k have word lengths k or k−1 and form a basis of Eλ_k. Here Eλ_k denotes the eigenspace to the eigenvalueλk.

Proof. — Letk∈2Z∩ {0, . . . , n}. By Lemma 5.4 and Lemma 5.5, we get that an eigenvectortⁱ₁¹. . . tⁱ_n−1ⁿ⁻¹ of word lengthk (herek6=n, ifn is even) with respect tot₁, . . . , t_n−1 corresponds to the eigenvalue

(−1)ⁱ¹+· · ·+ (−1)ⁱⁿ⁻¹+ (−1)ⁱ¹^+···+iⁿ⁻¹ =−k+ (n−1−k) + 1 =n−2k.

(17)

Now consider an eigenvectortⁱ₁¹. . . tⁱ_n−1ⁿ⁻¹ of word lengthk−1. Then we get the eigenvalue

(−1)ⁱ¹+· · ·+ (−1)ⁱⁿ⁻¹+ (−1)ⁱ¹^+···+iⁿ⁻¹

=−(k−1) + (n−k)−1 =n−2k.

We go through all the eigenvectors of Lemma 5.5 in this way and we obtain exactly the eigenvalues λk =n−2k. Since the eigenvectors of word lengthskork−1 are exactly those corresponding toλk, they form a basis

ofE_λ_k.

5.4. The quantum automorphism group ofF Q2m+1

For the rest of this section, we restrict to the foldedn-cube graphs, where n= 2m+ 1 is odd. We show that in this case, the quantum automorphism group isSO_n⁻¹. We need the following lemma.

Lemma 5.7. — Let τ1, . . . , τn be generators of C^∗(Zⁿ⁻¹2 ) with τ_i² = 1, τ_iτ_j = τ_jτ_i, τ_n = τ₁. . . τ_n−1 and let A be a C^∗-algebra with elements uij ∈ A fulfilling Relations (4.1)–(4.4). Let (i1, . . . , il) ∈ {1, . . . , n}^l with i_a6=i_b fora6=b, where16l6n. Then

n

X

j1,...,jl=1

τj₁. . . τj_l⊗uj₁i₁. . . uj_li_l= X

j1,...,jl; ja6=jbfora6=b

τj₁. . . τj_l⊗uj₁i₁. . . uj_li_l.

Proof. — Letjs=js+1=k and let the remainingjl be arbitrary. Sum- ming overk, we get

n

X

k=1

τj1. . . τjs−1τ_k²τjs+2. . . τj_l⊗uj1i1. . . ukisukis+1. . . uj_li_l

=τ_j₁. . . τ_j_s−1τ_j_s+2. . . τ_j_l⊗u_j₁_i₁. . .

n

X

k=1

u_ki_su_ki_s+1

! . . . u_j_l_i_l

= 0

by Relation (4.2) sinceis6=is+1. Doing this for alls∈ {1, . . . , l−1}we get

n

X

j1,...,j_l=1

τj₁. . . τj_l⊗uj₁i₁. . . uj_li_l = X

j₁6=···6=jl

τj₁. . . τj_l⊗uj₁i₁. . . uj_li_l.