*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 be a compact matrix quantum group and let be a . An (left) *action *of on is a unital *-homomorphism that satisfies the analogue of , and the Podlés density condition:

.

## Quantum Automorphism Groups of Finite Graphs

Schmidt in this earlier paper gives a slightly different presentation of . The definition given here I understand:

### Definition 1.3

The *quantum automorphism group* of a finite graph with adjacency matrix is given by the universal -algebra generated by such that the rows and columns of are partitions of unity and:

.

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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 by the ideal given by … ah but this is more or less the definition of universal -algebras given by generators and relations :

where presumably all works out OK, and it can be shown that 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 via the surjective *-homomorphism:

.

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## Compact Matrix Quantum Groups acting on Graphs

### Definition 1.6

Let be a finite graph and a compact matrix quantum group. An action of on is an action of on (coaction of on ) such that the associated magic unitary , given by:

,

commutes with the adjacency matrix, .

By the universal property, we have via the surjective *-homomorphism:

, .

## Theorem 1.8 (Banica)

Let , and , be an action, and let be a linear subspace given by a subset . The matrix commutes with the projection onto if and only if

### Corollary 1.9

The action preserves the eigenspaces of :

*Proof: *Spectral decomposition yields that each , or rather the projection onto it, satisfies a polynomial in :

,

as commutes with powers of

# A Criterion for a Graph to have Quantum Symmetry

### Definition 2.1

Let . Permutations are *disjoint *if , and vice versa, for all .

In other words, we don’t have and permuting any vertex.

### Theorem 2.2

Let be a finite graph. If there exists two non-trivial, disjoint automorphisms , such that and , then we get a surjective *-homomorphism . In this case, we have the quantum group , and so has quantum symmetry.

*Proof: *Suppose we have disjoing with and .

The group algebra of can be given as a universal -algebra (related to something I am looking at, the relationship between universal -algebras and simplicity, note that is simple under suitable assumptions… this is about when a concrete -algebra is isomorphic to a univeral one, or when two universal -algebras are isomorphic).

Anyway, we have:

The proof plans to use the universal property (of ) to get a surjective *-homormorphism onto , and as the do not commute, this gives quantum symmetries.

Identifying with their permutation matrices, define:

.

Schmidt shows that these satisfies the relations of 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

### Remark 2.3

Consider so that and . The disjoint automorpshisms gives the famous surjective *-homorphism exhibiting the quantum nature of . This subgroup is therefore isomorphic to, well its algebra of functions, to .

### Remark 2.4

Any pair of disjoint automorphisms give quantum symmetry,

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