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New Preprint: Tracing the orbitals of the quantum permutation group

February 11, 2023 in C*-Algebras & Operator Theory, Research, Research Updates | Leave a comment

Abstract: Using a suitably non-commutative flat matrix model, it is shown that the quantum permutation group has free orbitals: that is, a monomial in the generators of the algebra of functions can be zero for trivial reasons only. It is shown that any strict intermediate quantum subgroup between the classical and quantum permutation groups must have free three orbitals, and this is used to derive some elementary bounds for the Haar state on degree four monomials in such quantum permutation groups.

Link to arXiv

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New Preprint: Analysis for idempotent states on quantum permutation groups

January 31, 2023 in C*-Algebras & Operator Theory, Quantum Groups, Research, Research Updates | Leave a comment

Abstract: Woronowicz proved the existence of the Haar state for compact quantum groups under a separability assumption later removed by Van Daele in a new existence proof. A minor adaptation of Van Daele’s proof yields an idempotent state in any non-empty weak*-compact convolution-closed convex subset of the state space. Such subsets, and their associated idempotent states, are studied in the case of quantum permutation groups.

Link to arxiv.

Talk: The Kawada-Ito theorem for quantum groups

October 10, 2022 in C*-Algebras & Operator Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | Leave a comment

BCRI Mini-Symposium: Noncommutative Probability & Quantum Information

Monday, 10th 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.

kitv1Download

Talk: The Frucht property in the quantum group setting

January 25, 2022 in C*-Algebras & Operator Theory, Quantum Groups, Research, Research Updates | Leave a comment

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.

slidesDownload

A state-space approach to quantum permutations

January 13, 2022 in C*-Algebras & Operator Theory, Quantum Groups, Research, Research Updates | Leave a comment

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.

The Frucht property in the quantum group setting

November 19, 2021 in C*-Algebras & Operator Theory, Quantum Groups, Research, Research Updates | Leave a comment

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.

A state-space approach to quantum permutations

April 6, 2021 in C*-Algebras & Operator Theory, Quantum Groups, Research, Research Updates | Leave a comment

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.

Talk: Quantum Group Seminar

August 21, 2020 in C*-Algebras & Operator Theory, Quantum Groups, Research | Leave a comment

Giving a talk 17:00, September 1 2020:

See here for more.

Slides.

A Sufficient Condition for Quantum Symmetry

May 16, 2020 in arXiv, Quantum Groups | Leave a comment

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.

_______________________________________

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.

Read the rest of this entry »

Imagining non-Kac Quantum Groups

May 15, 2020 in Quantum Groups | Leave a comment

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 G as the set of symmetries on some kind of space, given by a set X together with some additional structure D(X). The elements of G  act on X as bijections:

G \ni g:X\rightarrow X,

such that D(X)=D(g(X)), that is the structure of the space is invariant under g.

For example, consider the space (X_n,|X_n|), where the set is X_n=\{1,2,\dots,n\}, and the structure is the cardinality. Then the set of all of the bijections X_n\rightarrow X_n is a group called S_n.

A set of symmetries G, a group, comes with some structure of its own. The identity map e:X\rightarrow X, x\mapsto x is a symmetry. By transitivity, symmetries g,h\in G can be composed to form a new symmetry gh:=g\circ h\in G. Finally, as bijections, symmetries have inverses g^{-1}, g(x)\mapsto x.

Note that:

gg^{-1}=g^{-1}g=e\Rightarrow (g^{-1})^{-1}=g.

A group can carry additional structure, for example, compact groups carry a topology in which the composition G\times G\rightarrow G and inverse {}^{-1}:G\rightarrow G are continuous.

Algebra of Functions

Given a group G together with its structure, one can define an algebra A(G) of complex valued functions on G, such that the multiplication A(G)\times A(G)\rightarrow A(G) is given by a commutative pointwise multiplication, for s\in G:

(f_1f_2)(s)=f_1(s)f_2(s)=(f_2f_1)(s).

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 A(G) we can reconstruct G.

Usually the following transpose maps will be considered in the structure of A(G), for some tensor product \otimes_\alpha such that A(G\times G)\cong A(G)\otimes_\alpha A(G), and m:G\times G\rightarrow G, (g,h)\mapsto gh is the group multiplication:

\begin{aligned} \Delta: A(G)\rightarrow A(G)\otimes_{\alpha}A(G)&,\,f\mapsto f\circ m,\,\text{the comultiplication} \\ S: A(G)\rightarrow A(G)&,\, f\mapsto f\circ {}^{-1},\,\text{ the antipode} \\ \varepsilon: A(G)\rightarrow \mathbb{C}&,\, f\mapsto f\circ e,\,\text{ the counit} \end{aligned}

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 \Delta: A(G)\rightarrow A(G)\otimes_\alpha A(G) for some tensor product \otimes_\alpha (or perhaps only into a multiplier algebra M(A(G)\otimes_\alpha A(G))), but in general that structure alone can only give a quantum semigroup.

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

A quantum group G is given by an algebra of functions A(G) satisfying a set of axioms \Theta such that:

  • whenever A(G) is noncommutative, G is a virtual object,
  • every commutative algebra of functions satisfying \Theta is an algebra of functions on a set-of-points group, and
  • whenever commutative algebras of functions A(G_1)\cong_{\Theta} A(G_2), G_1\cong G_2 as set-of-points groups.

Read the rest of this entry »

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