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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, 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.

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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.

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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:

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

_______________________________________

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 »

The Ergodic Theorem for Random Walks on Finite Quantum Groups

April 2, 2020 in C*-Algebras & Operator Theory, Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | 1 comment

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.

Freslon’s Ergodic Theorem for (Finite) Group Algebras

December 17, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

In the hope of gleaning information for the study of aperiodic random walks on (finite) quantum groups, which I am struggling with here, by couching Freslon’s Proposition 3.2, in the language of Fagnola and Pellicer, and in the language of my (failed) attempts (see here, here, and here) to find the necessary and sufficient conditions for a random walk to be aperiodic. It will be necessary to extract the irreducibility and aperiodicity from Freslon’s rather ‘unilateral’ result.

Well… I have an inkling that because dual groups satisfy what I would call the condition of abelianness (under the ‘quantisation’ functor), all (quantum) subgroups are normal… this is probably an obvious thing to write down (although I must search the literature) to ensure it is indeed known (or is untrue?). Edit: Wang had it already, see the last proposition here.

Let $F(G)$ be a the algebra of functions on a finite classical (as opposed to quantum) group $G$ . This has the structure of both an algebra and a coalgebra, with an appropriate relationship between these two structures. By taking the dual, we get the group algebra, $\mathbb{C}G=:F(\widehat{G})$ . The dual of the pointwise-multiplication in $F(G)$ is a coproduct for the algebra of functions on the dual group $\widehat{G}$ … this is all well known stuff.

Recall that the set of probabilities on a finite quantum group is the set of states $M_p(G):=\mathcal{S}(F(G))$ , and this lives in the dual, and the dual of $F(\widehat{G})$ is $F(G)$ , and so probabilities on $\widehat{G}$ are functions on $G$ . To be positive is to be positive definite, and to be normalised to one is to have $u(\delta^e)=1$ .

The ‘simplicity’ of the coproduct,

$\Delta(\delta^g)=\delta^g\otimes\delta^g$ ,

means that for $u\in M_p(\widehat{G})$ ,

$(u\star u)(\delta^g)=(u\otimes u)\Delta(\delta^g)=u(\delta^g)^2$ ,

so that, inductively, $u^{\star k}$ is equal to the (pointwise-multiplication power) $u^k$ .

The Haar state on $\widehat{G}$ is equal to:

$\displaystyle \pi:=\int_{\hat{G}}:=\delta_e$ ,

and therefore necessary and sufficient conditions for the convergence of $u^{\star k}\rightarrow \pi$ is that $u$ is strict. It can be shown that for any $u\in M_p(G)$ that $|u(\delta^g)|\leq u(\delta^e)=1$ . Strictness is that this is a strict inequality for $g\neq e$ , in which case it is obvious that $u^{\star k}\rightarrow \delta_e$ .

Here is a finite version of Freslon’s result which holds for discrete groups.

Freslon’s Ergodic Theorem for (Finite) Group Algebras

Let $u\in M_p(\widehat{G})$ be a probability on the dual of finite group. The random walk generated by $u$ is ergodic if and only if $u$ is not-concentrated on a character on a non-trivial subgroup $H\subset G$ .

Freslon’s proof passes through the following equivalent condition:

The random walk on $\widehat{G}$ driven by $u\in M_p(\widehat{G})$ is not ergodic if $u$ is bimodularwith respect to a non-trivial subgroup $H\subset G$ , in the sense that

$\displaystyle u(\underbrace{\delta^g\delta^h}_{=\delta^{gh}})=u(\delta^g)u(\delta^h)=u\left(\underbrace{\delta^h\delta^g}_{=\delta^{hg}}\right)$ .

Before looking at the proof proper, we might note what happens when $G$ is abelian, in which case $\widehat{G}$ is a classical group, the set of characters on $G$ .

To every positive definite function $u\in M_p(\widehat{G})$ , we can associate a probability $\nu_u\in M_p(\widehat{G})$ such that:

$\displaystyle u(\delta^g)=\sum_{\chi\in\hat{G}} \chi(\delta^g) \nu_u(\chi)$ .

This is Bochner’s Theorem for finite abelian groups. This implies that positive definite functions on finite abelian groups are exactly convex combinations of characters.

Freslon’s condition says that to be not ergodic, $u$ must be a character on a non-trivial subgroup $H\subset G$ . Such characters can be extended in $[G\,:\,H]$ ways.

Therefore, if $u$ is not ergodic, $u_{\left|H\right.}=\eta\in \widehat{H}$ .

For $h\in H$ , we have

$\displaystyle u(h)=\sum_{\chi\in\widehat{G}}\chi(h)\nu_u(\chi)$ ,

dividing both sides by $u(h)=\eta(h)\neq 0$ yields:

$\displaystyle\sum_{\chi \in \widehat{G}} (\eta^{-1}\chi)(h)\nu_u (\chi)=1$ .

As $\nu_u\in M_p(\widehat{G})$ , and $(\eta^{-1}\chi)(h)\in \mathbb{T}$ , this implies that $\nu_u$ is supported on characters such that, for all $h\in H$ :

$\eta^{-1}(h)\chi(h)=1\Rightarrow \chi=\eta\tilde{\chi}$ ,

such that $\tilde{\chi}(H)=\{1\}$ . The set of such $\tilde{\chi}$ is the annihilator of $H$ in $\widehat{G}$ , and it is a subgroup. Therefore $\nu_u$ is concentrated on the coset of a normal subgroup (as all subgroups of an abelian group are normal).

This, via Pontragin duality, is not looking at the ‘support’ of $u$ , but rather of $\nu_u$ . Although we denote $\mathbb{C}G=:F(\widehat{G})$ , and when $G$ is abelian, $\widehat{G}$ is a group (unnaturally, of characters) isomorphic to $G$ . Is it the case though that,

$\Delta(\chi)=\chi\otimes\chi$

gives the same object in as

$\displaystyle\Delta(\chi)=\sum_{g\in G}\chi(\delta^g)\Delta(\delta_g)$

$\displaystyle =\sum_{g\in G}\chi(\delta^g)\sum_{t\in G}\delta_{gt^{-1}}\otimes \delta_t$ ?

Well… of course this is true because $\chi(gh)=\chi(g)\chi(h)$ .

We could proceed to look at Fagnola & Pellicer’s work but first let us prove Freslon’s result, hopefully in the finite case the analysis disappears…

Proof: Assume that $u$ is not strict and let

$\Lambda:=|u|^{-1}(\{1\})$ .

There exists a unitary representation $\Phi:G\rightarrow B(H)$ and a unit vector $\xi$ such that

$u(g)=\langle \Phi(g)\xi,\xi\rangle$

Cauchy-Schwarz implies that

$|u(g)|\leq \|\Phi(g)\xi\|\|\xi\|=\|\xi\|^2$ .

If $h$ is not strict there is an $h$ such that this is an inequality and so $\Phi(h)\xi$ is colinear to $\xi$ , it follows that $\Phi(h)\xi=u(h)\xi$ .

This implies for $h\in \Lambda$ and $g\in G$ :

$|u(gh)|=|\langle \Phi(gh)\xi,\xi\rangle|=|u(h)||\langle \Phi(g)\xi,\xi\rangle|=|u(g)|$ ,

and so $\Lambda$ is closed under multiplication. Also $u(g^{-1})=\overline{u(g)}$ and so $\Lambda$ and so $\Lambda$ is a subgroup. It follows that $u$ is a character on $\Lambda$ , which is not trivial because $u$ is not strict.

I don’t really need to go through the third equivalent condition. If $u$ coincides with a character on a subgroup $\Lambda$ , for $h\in \Lambda$

$|u(h)|^2=u(h)\overline{u(h)}=u(h)u(h^{-1})=u(e)=1$ ,

and so $u$ is not strict $\bullet$

Now let us look at the language of Fagnola and Pellicer. What is a projection in $\mathbb{C}G$ ? First note the involution in $\mathbb{C}G$ is $(\delta^g)^*=\delta^{g^{-1}}$ . The second multiplication is the convolution. This means projections in the algebra are symmetric with respect to the group inverses and they are also idempotents. They are actually equal to Haar states on finite subgroups.

I think periodicity is also wrapped up in Freslon’s result as I think all subgroups of dual groups are normal. Perhaps, oddly, not being concentrated on a subgroup means that the positive function (probability on the dual) is one on that subgroup…

Well… let us start with irreducible. Suppose $u$ fails to be ergodic because it is irreducible. This means there is a projection $p_H=\int_H$ such that that $P_u(p_H)=p_H$ (and support $u$ less than $p_H$ ?)

Let us look at the first condition:

$P_u(p_H)=(u\otimes I)\Delta(p_H)=\cdots=\frac{1}{|H|}\sum_{h\in H}u(h)\delta^h=p_H\Rightarrow u_{\left|H\right.}=1$ .

What now is the support of $u$ ? Well… some work I have done offline shows that the special projections, the group-like projections, correspond to to $\pi_H$ for $H$ a subgroup of $G$ . If $u$ is reducible, it is concentrated on such a quasi-subgroup, and this means that $u$ coincides with a trivial character on $H$ . In terms of Fagnola Pellicer, $P_u(\pi_H)=\pi_H$ .

Now let us tackle aperiodicity. It is going to correspond, I think, with being concentrated on a ‘coset’ of a non-trivial character on $H$ …

Well, we can show that if $u$ is periodic, there is a subset $S\subset G$ such that $u(s)=e^{2\pi i a_s/d}$ for all $s\in S$ . We can use Freslon’s proof to show that $S$ is in a subgroup on which $|u|=1$ .

Now what I want to do is put this in the language of ‘inclusion’ matrices… but the inclusions for cocommutative quantum groups are trivial so no go…

We can reduce Freslon’s conditions down to irreducible and aperiodic: not coinciding with a trivial character, and not coinciding with a character.

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Category Archive

Talk: The Kawada-Ito theorem for quantum groups

Talk: The Frucht property in the quantum group setting

A state-space approach to quantum permutations

Abstract

The Frucht property in the quantum group setting

Abstract

A state-space approach to quantum permutations

Abstract

Talk: Quantum Group Seminar

A Sufficient Condition for Quantum Symmetry

Preliminaries

Compact Matrix Quantum Groups

Definition 1.2

Quantum Automorphism Groups of Finite Graphs

Definition 1.3

Compact Matrix Quantum Groups acting on Graphs

Definition 1.6

Theorem 1.8 (Banica)

Corollary 1.9

A Criterion for a Graph to have Quantum Symmetry

Definition 2.1

Theorem 2.2

Imagining non-Kac Quantum Groups

Groups

Algebra of Functions

Quantum Groups

The Ergodic Theorem for Random Walks on Finite Quantum Groups

Abstract

Freslon’s Ergodic Theorem for (Finite) Group Algebras

Freslon’s Ergodic Theorem for (Finite) Group Algebras

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Preliminaries

Compact Matrix Quantum Groups

Definition 1.2

Quantum Automorphism Groups of Finite Graphs

Definition 1.3

Compact Matrix Quantum Groups acting on Graphs

Definition 1.6

Theorem 1.8 (Banica)

Corollary 1.9

A Criterion for a Graph to have Quantum Symmetry

Definition 2.1

Theorem 2.2

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Groups

Algebra of Functions

Quantum Groups

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Abstract

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Freslon’s Ergodic Theorem for (Finite) Group Algebras

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