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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.
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 semigroup.
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 virtual object,
- every commutative algebra of functions satisfying
is an algebra of functions on a set-of-points group, and
- whenever commutative algebras of functions
,
as set-of-points groups.
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.
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 be a the algebra of functions on a finite classical (as opposed to quantum) group
. 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,
. The dual of the pointwise-multiplication in
is a coproduct for the algebra of functions on the dual group
… this is all well known stuff.
Recall that the set of probabilities on a finite quantum group is the set of states , and this lives in the dual, and the dual of
is
, and so probabilities on
are functions on
. To be positive is to be positive definite, and to be normalised to one is to have
.
The ‘simplicity’ of the coproduct,
,
means that for ,
,
so that, inductively, is equal to the (pointwise-multiplication power)
.
The Haar state on is equal to:
,
and therefore necessary and sufficient conditions for the convergence of is that
is strict. It can be shown that for any
that
. Strictness is that this is a strict inequality for
, in which case it is obvious that
.
Here is a finite version of Freslon’s result which holds for discrete groups.
Freslon’s Ergodic Theorem for (Finite) Group Algebras
Let be a probability on the dual of finite group. The random walk generated by
is ergodic if and only if
is not-concentrated on a character on a non-trivial subgroup
.
Freslon’s proof passes through the following equivalent condition:
The random walk on driven by
is not ergodic if
is bimodularwith respect to a non-trivial subgroup
, in the sense that
.
Before looking at the proof proper, we might note what happens when is abelian, in which case
is a classical group, the set of characters on
.
To every positive definite function , we can associate a probability
such that:
.
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, must be a character on a non-trivial subgroup
. Such characters can be extended in
ways.
Therefore, if is not ergodic,
.
For , we have
,
dividing both sides by yields:
.
As , and
, this implies that
is supported on characters such that, for all
:
,
such that . The set of such
is the annihilator of
in
, and it is a subgroup. Therefore
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 , but rather of
. Although we denote
, and when
is abelian,
is a group (unnaturally, of characters) isomorphic to
. Is it the case though that,
gives the same object in as
?
Well… of course this is true because .
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 is not strict and let
.
There exists a unitary representation and a unit vector
such that
Cauchy-Schwarz implies that
.
If is not strict there is an
such that this is an inequality and so
is colinear to
, it follows that
.
This implies for and
:
,
and so is closed under multiplication. Also
and so
and so
is a subgroup. It follows that
is a character on
, which is not trivial because
is not strict.
I don’t really need to go through the third equivalent condition. If coincides with a character on a subgroup
, for
,
and so is not strict
Now let us look at the language of Fagnola and Pellicer. What is a projection in ? First note the involution in
is
. 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 fails to be ergodic because it is irreducible. This means there is a projection
such that that
(and support
less than
?)
Let us look at the first condition:
.
What now is the support of ? Well… some work I have done offline shows that the special projections, the group-like projections, correspond to to
for
a subgroup of
. If
is reducible, it is concentrated on such a quasi-subgroup, and this means that
coincides with a trivial character on
. In terms of Fagnola Pellicer,
.
Now let us tackle aperiodicity. It is going to correspond, I think, with being concentrated on a ‘coset’ of a non-trivial character on …
Well, we can show that if is periodic, there is a subset
such that
for all
. We can use Freslon’s proof to show that
is in a subgroup on which
.
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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