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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.

### 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.

In my pursuit of an Ergodic Theorem for Random Walks on (probably finite) Quantum Groups, I have been looking at analogues of Irreducible and Periodic. I have, more or less, got a handle on irreducibility, but I am better at periodicity than aperiodicity.

The question of how to generalise these notions from the (finite) classical to noncommutative world has already been considered in a paper (whose title is the title of this post) of Fagnola and Pellicer. I can use their definition of periodic, and show that the definition of irreducible that I use is equivalent. This post is based largely on that paper.

## Introduction

Consider a random walk on a finite group driven by . The state of the random walk after steps is given by , defined inductively (on the algebra of functions level) by the associative

.

The convolution is also implemented by right multiplication by the stochastic operator:

,

where has entries, with respect to a basis . Furthermore, therefore

,

and so the stochastic operator describes the random walk just as well as the driving probabilty .

The random walk driven by is said to be *irreducible *if for all , there exists such that (if ) .

The *period *of the random walk is defined by:

.

The random walk is said to be *aperiodic *if the period of the random walk is one.

These statements have counterparts on the set level.

If is not irreducible, there exists a proper subset of , say , such that the set of functions supported on are -invariant. It turns out that is a proper subgroup of .

Moreover, when is irreducible, the period is the greatest common divisor of all the natural numbers such that there exists a partition of such that the subalgebras of functions supported in satisfy:

and (slight typo in the paper here).

In fact, in this case it is necessarily the case that is concentrated on a coset of a proper normal subgroup , say . Then .

Suppose that is supported on . We want to show that for . Recall that

.

This shows how the stochastic operator reduces the index .

A central component of Fagnola and Pellicer’s paper are results about how the decomposition of a stochastic operator:

,

specifically the maps can speak to the irreducibility and periodicity of the random walk given by . I am not convinced that I need these results (even though I show how they are applicable).

## Stochastic Operators and Operator Algebras

Let be a -algebra (so that is in general a virtual object). A -subalgebra is *hereditary *if whenever and , and , then .

It can be shown that if is a hereditary subalgebra of that there exists a projection such that:

.

All hereditary subalgebras are of this form.

*This sandbox is going to take from a variety of sources, mostly Shuzhou Wang.*

## C*-Ideals

Let be a closed (two-sided) ideal in a non-commutative unital -algebra . Such an ideal is self-adjoint and so a non-commutative -algebra . The quotient map is given by , , where is the equivalence class of under the equivalence relation:

.

Where we have the product

,

and the norm is given by:

,

the quotient is a -algebra.

Consider now elements and . Consider

.

The tensor product . Now note that

,

by the nature of the Tensor Product (). Therefore .

### Definition

A WC*-ideal (W for *Woronowicz*) is a C*-ideal such that , where is the quotient map .

Let be the algebra of functions on a classical group . Let . Let be the set of functions which vanish on : this is a C*-ideal. The kernal of is .

Let so that . Note that

and so

.

Note that if . It is not possible that both and are in : if they were , but , which is not in by assumption. Therefore one of or is equal to zero and so:

,

and so by linearity, if vanishes on a subgroup ,

.

In this way, WC*-ideals generalise functions which vanish on distinguished subgroups. In fact, without checking all the details, I imagine that first isomorphism theorem can show that . Let be the ring homomorphism

.

Then , , and so we have the isomorphism of rings, which presumably carries forward to the algebras of functions level…

## Distances between Probability Measures

Let be a finite quantum group and be the set of states on the -algebra .

The algebra has an invariant state , the dual space of .

Define a (bijective) map , by

,

for .

Then, where and , define the total variation distance between states by

.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative machinary shows that:

.

(supremum presentation)

In the classical case, using the test function , where , we have the probabilists’ preferred definition of total variation distance:

.

In the classical case the set of indicator functions on the subsets of the group exhaust the set of projections in , and therefore the classical total variation distance is equal to:

.

(Projection Distance)

In all cases the quantum total variation distance and the supremum presentation are equal. In the classical case they are equal also to the projection distance. Therefore, in the classical case, we are free to define the total variation distance by the projection distance.

## Quantum Projection Distance Quantum Variation Distance?

Perhaps, however, on truly quantum finite groups the projection distance could differ from the QTVD. In particular, a pair of states on a factor of might be different in QTVD vs in projection distance (this cannot occur in the classical case as all the factors are one dimensional).

*Taken from C*-Algebras and Operator Theory by Gerald J. Murphy.*

Although the principal aim of this section is to construct direct limits of C*-algebras, we begin with direct limits of groups.

If is a sequence of groups, and if for each we have a homomorphism , then we call a *direct sequence of groups. *Given such a sequence and positive integers , we set and we define inductively on by setting

.

If , we have .

If is a group and we have homomorphisms such that the diagram

commutes for each , that is , then for all .

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