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**BCRI Mini-Symposium**: **Noncommutative Probability & Quantum Information**

**Monday, 10 ^{th} October 2022 from 12:00 to 15:00**

**Organizers: Claus Koestler (UCC), Stephen Wills (UCC)**

**SPEAKER: **J.P. McCarthy (Munster Technological University)**TITLE:** The Kawada-Itô theorem for finite quantum groups.**ABSTRACT:** *Necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This result is manifest in the case of random walks on finite groups by a statement about the support of the driving probability: a random walk on a finite group is ergodic if and only if the support is not concentrated on a proper subgroup, nor on a coset of a proper normal subgroup. The study of random walks on finite groups extends naturally to the study of random walks on compact quantum groups, where a state on the algebra of functions plays the role of the driving probability. A random walk on a compact quantum group can fail to be irreducible without being concentrated on a proper quantum subgroup. In this talk we will explore this phenomenon. Time allowing, we will talk about periodicity, and as a conclusion, I give necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group in terms of the support projection of the driving state.*

In the end the talk (below) didn’t quite match the abstract.

Quantum Group Seminar, Monday 24 January, 2022.

**Abstract**: A classical theorem of Frucht states that every finite group is the automorphism group of a finite graph. Is every quantum permutation group the quantum automorphism group of a finite graph? In this talk we will answer this question with the help of orbits and orbitals.

This talk is based on joint work with Teo Banica.

This article has been accepted to Expositiones Mathematicae.

### Abstract

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an *element* of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.

Anyone clicking here before March 04, 2022 will be taken directly to the latest version of the article on ScienceDirect, which they are welcome to read or download. No sign up, registration or fees are required.

### Abstract

A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.

with Teo Banica, Glasgow Math J., to appear. Arxiv link here.

### Abstract

An exposition of quantum permutation groups where an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and posits that states on the algebra of continuous functions on a quantum permutation group can be interpreted as quantum permutations. This interpretation allows talk of an element of a compact quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated with the Kac-Paljutkin quantum group, the duals of finite groups, as well as by other finite quantum group phenomena.

Arxiv link here.

*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:

.

_______________________________________

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:

.

_______________________________________

## 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.

*Warning: This is written by a non-expert (I know only about finite quantum groups and am beginning to learn my compact quantum groups), and there is no attempt at rigour, or even consistency. Actually the post shows a wanton disregard for reason, and attempts to understand the incomprehensible and intuit the non-intuitive. Speculation would be too weak an adjective.*

## Groups

A group is a well-established object in the study of mathematics, and for the purposes of this post we can think of a group as the set of symmetries on some kind of space, given by a set together with some additional structure . The elements of *act *on as bijections:

,

such that , that is the structure of the space is invariant under .

For example, consider the space , where the set is , and the structure is the cardinality. Then the set of all of the bijections is a group called .

A set of symmetries , a group, comes with some structure of its own. The identity map , is a symmetry. By transitivity, symmetries can be composed to form a new symmetry . Finally, as bijections, symmetries have inverses , .

Note that:

.

A group can carry additional structure, for example, compact groups carry a topology in which the composition and inverse are continuous.

## Algebra of Functions

Given a group together with its structure, one can define an algebra of complex valued functions on , such that the multiplication is given by a commutative pointwise multiplication, for :

.

Depending on the class of group (e.g. finite, matrix, compact, locally compact, etc.), there may be various choices and considerations for what algebra of functions to consider, but on the whole it is nice if given an algebra of functions we can reconstruct .

Usually the following *transpose *maps will be considered in the structure of , for some tensor product such that , and , is the group multiplication:

See Section 2.2 to learn more about these maps and the relations between them for the case of the complex valued functions on *finite *groups.

## Quantum Groups

Quantum groups, famously, do not have a single definition in the same way that groups do. All definitions I know about include a coassociative (see Section 2.2) comultiplication for some tensor product (or perhaps only into a multiplier algebra ), but in general that structure alone can only give a quantum *semi*group.

Here is a non-working (quickly broken?), meta-definition, inspired in the usual way by the famous Gelfand Theorem:

A quantum group is given by an algebra of functions satisfying a set of axioms such that:

- whenever is noncommutative, is a
virtualobject,every commutative algebra of functions satisfying is an algebra of functions on aset-of-pointsgroup, andwhenever commutative algebras of functions , asset-of-pointsgroups.

Some notes on this paper.

### 1. Introduction and Main Results

A tree *has no symmetry* if its automorphism group is trivial. Erdos and Rényi showed that the probability that a random tree on vertices *has no symmetry *goes to zero as .

Banica (after Bichon) wrote down with clarity the quantum automorphism group of a graph. It contains the usual automorphism group. When it is larger, the graph is said to have *quantum symmetry*.

Lupini, Mancinska, and Roberson show that almost all graphs are quantum antisymmetric. I am fairly sure this means that almost all graphs have no quantum symmetry, and furthermore for almost all (as ) graphs the automorphism group is trivial.

The paper in question hopes to show that almost all trees have quantum symmetry — but at this point I am not sure if this is saying that the quantum automorphism group is larger than the classical.

## 2. Preliminaries

### 2.1 Graphs and Trees

Standard definitions. No multi-edges. *Undirected* if the edge relation is symmetric. As it is dealing with trees, this paper is concerned with undirected graphs without loops, and identify . A *path *is a sequence of edges. We will not see *cycles *if we are discussing trees. Neither will we talk about *disconnected* graphs: a *tree *is a connected graph without cycles (this throws out loops actually.

The *adjacency matrix *of a graph is a matrix with iff there is an edge connected and . The adjacency matrix is symmetric.

### 2.2 Symmetries of Graphs

An *automorphism *of a graph is a permutation of that preserves adjacency and non-adjacency. The set of all such automorphisms, , is a group where the group law is composition. It is a subgroup of , and itself can be embedded as permutation matrices in . We then have

.

If , it is asymmetric. Otherwise it is or rather has symmetry.

### 2.3 Compact Matrix Quantum Groups

A *compact matrix quantum group *is a pair , where is a unital -algebra, and is such that:

- is generated by the ,
- There exists a morphism , such that
- and are invertible (Timmermann only asks that be invertible)

The classic example (indeed commutative examples all take this form) is a compact matrix group and the coordinates of .

#### Example 2.3

The algebra of continuous functions on the quantum permutation group is generated by projections such that the row sums and column sums of both equal .

The map , is a surjective morphism that is an isomorphism for , so that the sets have no quantum symmetries.

### 2.4 Quantum Symmetries of Graphs

#### Definition 2.4 (Banica after Bichon)

Let be a graph on vertices without multiple edges not loops, and let be its adjacency matrix. The *quantum automorphism group* is defined as the compact matrix group with -algebra:

For me, not the authors, this requires some work. Banica says that is a *Hopf ideal*.

A *Hopf ideal *is a closed *-ideal such that

.

Classically, the set of functions vanishing on a distinguished subgroup. The quotient map is , and if their difference is in , that is if they agree on the subgroup.

The classical version of ends up as … the group in question the classical . In that sense perhaps might be better given as .

Easiest thing first, is it a *-ideal? Well, take the adjoint of and so is *closed. Suppose and … I cannot prove that this is an ideal! But time to move on.

### 3. The Existence of Two Cherries

In this section the authors will show that almost all trees have two cherries. Definition 3.4 says with clarity what a *cherry *is, here I use an image [credit: www-math.ucdenver.edu]:

*(3,5,4) and (7,9,8) are cherries*

#### Remark 3.2

If a graph admits a cherry , the transposition is a non-trivial automorphism.

#### Theorem 3.3 (Erdos, Réyni)

*Almost all trees contains at least one cherry in the sense that*

,

*where is #cherries in a (uniformly chosen) random tree on vertices.*

#### Corollary 4.3

*Almost all trees have symmetry. *

The paper claims in fact that almost all trees have at least two cherries. This will allow some action to take place. This can be seen in this paper which is the next point of interest.

### Abstract

Necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This result is manifest in the case of random walks on finite groups by a statement about the support of the driving probability: a random walk on a finite group is ergodic if and only if the support is not concentrated on a proper subgroup, nor on a coset of a proper normal subgroup. The study of random walks on finite groups extends naturally to the study of random walks on finite quantum groups, where a state on the algebra of functions plays the role of the driving probability. Necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group are given on the support projection of the driving state.

Link to journal here.

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