This post follows on from this one. The purpose of posts in this category is for me to learn more about the research being done in quantum groups. This post looks at this paper of Schmidt.

# Preliminaries

## Compact Matrix Quantum Groups

The author gives the definition and gives the definition of a (left, quantum) group action.

### Definition 1.2

Let $G$ be a compact matrix quantum group and let $C(X)$ be a $\mathrm{C}^*-algebra$. An (left) action of $G$ on $X$ is a unital *-homomorphism $\alpha: C(X)\rightarrow C(X)\otimes C(G)$ that satisfies the analogue of $g_2(g_1x)=(g_2g_1)x$, and the Podlés density condition:

$\displaystyle \overline{\text{span}(\alpha(C(X)))(\mathbf{1}_X\otimes C(G))}=C(X)\otimes C(G)$.

## Quantum Automorphism Groups of Finite Graphs

Schmidt in this earlier paper gives a slightly different presentation of $\text{QAut }\Gamma$. The definition given here I understand:

### Definition 1.3

The quantum automorphism group of a finite graph $\Gamma=(V,E)$ with adjacency matrix $A$ is given by the universal $\mathrm{C}^*$-algebra $C(\text{QAut }\Gamma)$ generated by $u\in M_n(C(\text{QAut }\Gamma))$ such that the rows and columns of $u$ are partitions of unity and:

$uA=Au$.

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The difference between this definition and the one given in the subsequent paper is that in the subsequent paper the quantum automorphism group is given as a quotient of $C(S_n^+)$ by the ideal given by $\mathcal{I}=\langle Au=uA\rangle$… ah but this is more or less the definition of universal $\mathrm{C}^*$-algebras given by generators $E$ and relations $R$:

$\displaystyle \mathrm{C}^*(E,R)=\mathrm{C}^*( E)/\langle \mathcal{R}\rangle$

$\displaystyle \Rightarrow \mathrm{C}^*(E,R)/\langle I\rangle=\left(\mathrm{C}^*(E)/R\right)/\langle I\rangle=\mathrm{C}*(E)/(\langle R\rangle\cap\langle I\rangle)=\mathrm{C}^*(E,R\cap I)$

where presumably $\langle R\rangle \cap \langle I \rangle=\langle R\cap I\rangle$ all works out OK, and it can be shown that $I$ is a suitable ideal, a Hopf ideal. I don’t know how it took me so long to figure that out… Presumably the point of quotienting by (a presumably Hopf) ideal is so that the quotient gives a subgroup, in this case $\text{QAut }\Gamma\leq S_{|V|}^+$ via the surjective *-homomorphism:

$C(S_n^+)\rightarrow C(S_n^+)/\langle uA=Au\rangle=C(\text{QAut }\Gamma)$.

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## Compact Matrix Quantum Groups acting on Graphs

### Definition 1.6

Let $\Gamma$ be a finite graph and $G$ a compact matrix quantum group. An action of $G$ on $\Gamma$ is an action of $G$ on $V$ (coaction of $C(G)$ on $C(V)$) such that the associated magic unitary $v=(v_{ij})_{i,j=1,\dots,|V|}$, given by:

$\displaystyle \alpha(\delta_j)=\sum_{i=1}^{|V|} \delta_i\otimes v_{ij}$,

commutes with the adjacency matrix, $uA=Au$.

By the universal property, we have $G\leq \text{QAut }\Gamma$ via the surjective *-homomorphism:

$C(\text{QAut }\Gamma)\rightarrow C(G)$, $u\mapsto v$.

### Theorem 1.8 (Banica)

Let $X_n=\{1,\dots,n\}$, and $\alpha:F(X_n)\rightarrow F(X_n)\otimes C(G)$, $\alpha(\delta_j)=\sum_i\delta_i\otimes v_{ij}$ be an action, and let $F(K)$ be a linear subspace given by a subset $K\subset X_n$. The matrix $v$ commutes with the projection onto $F(K)$ if and only if $\alpha(F(K))\subseteq F(K)\otimes C(G)$

### Corollary 1.9

The action $\alpha: F(V)\rightarrow F(V)\otimes C(\text{QAut }\Gamma)$ preserves the eigenspaces of $A$:

$\alpha(E_\lambda)\subseteq E_\lambda\otimes C(\text{QAut }\Gamma)$

Proof: Spectral decomposition yields that each $E_\lambda$, or rather the projection $P_\lambda$ onto it, satisfies a polynomial in $A$:

$\displaystyle P_\lambda=\sum_{i}c_iA^i$

$\displaystyle \Rightarrow P_\lambda A=\left(\sum_i c_i A^i\right)A=A P_\lambda$,

as $A$ commutes with powers of $A$ $\qquad \bullet$

# A Criterion for a Graph to have Quantum Symmetry

### Definition 2.1

Let $V=\{1,\dots,|V|\}$. Permutations $\sigma,\,\tau: V\rightarrow V$ are disjoint if $\sigma(i)\neq i\Rightarrow \tau(i)=i$, and vice versa, for all $i\in V$.

In other words, we don’t have $\sigma$ and $\tau$ permuting any vertex.

### Theorem 2.2

Let $\Gamma$ be a finite graph. If there exists two non-trivial, disjoint automorphisms $\sigma,\tau\in\text{Aut }\Gamma$, such that $o(\sigma)=n$ and $o(\tau)=m$, then we get a surjective *-homomorphism $C(\text{QAut }\Gamma)\rightarrow C^*(\mathbb{Z}_n\ast \mathbb{Z}_m)$. In this case, we have the quantum group $\widehat{\mathbb{Z}_n\ast \mathbb{Z}_m}\leq \text{QAut }\Gamma$, and so $\Gamma$ has quantum symmetry.

Proof: Suppose we have disjoing $\sigma,\,tau$ with $o(\sigma)=n$ and $o(\tau)=m$.

The group algebra of $\mathbb{Z}_n\ast \mathbb{Z}_m$ can be given as a universal $\mathrm{C}^*$-algebra (related to something I am looking at, the relationship between universal $\mathrm{C}^*$-algebras and simplicity, note that $\mathrm{C}^*_r(G_1\ast G2)$ is simple under suitable assumptions… this is about when a concrete $\mathrm{C}^*$-algebra is isomorphic to a univeral one, or when two universal $\mathrm{C}^*$-algebras are isomorphic).

Anyway, we have:

$\displaystyle \mathrm{C}^*(\mathbb{Z}_n \ast \mathbb{Z}_m)=\mathrm{C}^*(\{p_i\}_{i=1,\dots,n},\,\{q_i\}_{j=1,\dots,m}\,:\, \{p_i\}_{i=1,\dots,n},\,\{q_i\}_{j=1,\dots,m}\text{ partitions of unity})$

The proof plans to use the universal property (of $C(\text{QAut }\Gamma)$) to get a surjective *-homormorphism onto $\mathrm{C}^*(\mathbb{Z}_n\ast \mathbb{Z}_m)$, and as the $p_i,\,q_i$ do not commute, this gives quantum symmetries.

Identifying $\sigma,\,\tau\in\text{Aut }\Gamma$ with their permutation matrices, define:

$\displaystyle u':=\sum_{\ell=1}^m\tau^\ell \otimes q_\ell +\sum_{k=1}^n\sigma^k\otimes p_k-I_{M_{|V|}(\mathbb{C})}\otimes \mathbf{1}_{\widehat{\mathbb{Z}_n\ast \mathbb{Z}_m}}\in M_{|V|}(\mathbb{C})\otimes \mathrm{C}^*(\mathbb{Z}_n\ast \mathbb{Z}_m)\cong M_{|V|}(\mathrm{C}^*(\mathbb{Z}_n\ast \mathbb{Z}_m))$.

Schmidt shows that these $u'_{ij}$ satisfies the relations of $\text{QAut }\Gamma$ and thus we have a *-homormophism.

I thought such maps were automatically surjective… but obviously not. Schmidt shows that it is and so the rest follows $\qquad\bullet$

### Remark 2.3

Consider $K_4$ so that $\text{Aut }K_4=S_4$ and $\text{QAut }K_4=S_4^+$. The disjoint automorpshisms $\sigma=(1\quad 2),\,\tau=(3\qquad 4)\in S_4$ gives the famous surjective *-homorphism exhibiting the quantum nature of $S_4^+$. This subgroup is therefore isomorphic to, well its algebra of functions, to $\mathrm{C}*(\mathbb{Z}_2\ast \mathbb{Z}_2)$.

### Remark 2.4

Any pair of disjoint automorphisms give quantum symmetry,