You are currently browsing the category archive for the ‘Research’ category.
I am currently (slowly) working on an essay/paper where I expand upon the ideas in this talk. In this post I will try and explain in this framework why there is no quantum cyclic group, no quantum , and ask why there is no quantum alternating group.
Quantum Permutations Basics
Let be a unital
-algebra. We say that a matrix
is a magic unitary if each entry is a projection
, and each row and column of
is a partition of unity, that is:
.
It is necessarily the case (but not for *-algebras) that elements along the same row or column are orthogonal:
and
.
Shuzou Wang defined the algebra of continuous functions on the quantum permutation group on symbols to be the universal
-algebra
generated by an
magic unitary
. Together with (leaning heavily on the universal property) the *-homomorphism:
,
and the fact that and
are invertible (
), the quantum permutation group
is a compact matrix quantum group.
Any compact matrix quantum group generated by a magic unitary is a quantum permutation group in that it is a quantum subgroup of the quantum permutation group. There are finite quantum groups (finite dimensional algebra of functions) which are not quantum permutation groups and so Cayley’s Theorem does not hold for quantum groups. I think this is because we can have quantum groups which act on algebras such as rather than
— the algebra of functions equivalent of the finite set
.
This is all basic for quantum group theorists and probably unmotivated for everyone else. There are traditional motivations as to why such objects should be considered algebras of functions on quantum groups:
- find a presentation of an algebra of continuous functions on a group,
, as a commutative universal
-algebra. Study the the same object liberated by dropping commutativity. Call this the quantum or free version of
,
.
- quotient
by the commutator ideal, that is we look at the commutative
algebra generated by an
magic unitary. It is isomorphic to
, the algebra of functions on (classical)
.
- every commutative algebra of continuous functions on a compact matrix quantum group is the algebra of functions on a (classical) compact matrix group, etc.
Here I want to take a very different direction which while motivationally rich might be mathematically poor.
Weaver Philosophy
Take a quantum permutation group and represent the algebra of functions as bounded operators on a Hilbert space
. Consider a norm-one element
as a quantum permutation. We study the properties of the quantum permutation by making a series of measurements using self-adjoint elements of
.
Suppose we have a finite-spectrum, self-adjoint measurement . It’s spectral decomposition gives a partition of unity
. The measurement of
with
gives the value
with probability:
,
and we have the expectation:
.
What happens if the measurement of with
yields
(which can only happen if
)? Then we have some wavefunction collapse of
.
Now we can keep playing the game by taking further measurements. Notationally it is easier to describe what is happening if we work with projections (but straightforward to see what happens with finite-spectrum measurements). At this point let me quote from the essay/paper under preparation:
Suppose that the “event” has been observed so that the state is now
. Note this is only possible if
is non-null in the sense that
The probability that measurement produces , and
, is:
Define now the event , said “given the state
,
is measured to be
after
is measured to be
“. Assuming that
is non-null, using the expression above a probability can be ascribed to this event:
Inductively, for a finite number of projections , and
:
In general, and so
and this helps interpret that and
are not simultaneously observable. However the sequential projection measurement
is “observable” in the sense that it resembles random variables with values in
. Inductively the sequential projection measurement
resembles a
-valued random variable, and
If and
do commute, they share an orthonormal eigenbasis, and it can be interpreted that they can “agree” on what they “see” when they “look” at
, and can thus be determined simultaneously. Alternatively, if they commute then the distributions of
and
are equal in the sense that
it doesn’t matter what order they are measured in, the outputs of the measurements can be multiplied together, and this observable can be called .
Consider the (classical) permutation group or moreover its algebra of functions
. The elements of
can be represented as bounded operators on
, and the algebra is generated by a magic unitary
where:
.
Here (‘unrepresented’) that asks of
… do you send
? One for yes, zero for no.
Recall that the product of commuting projections is a projection, and so as is commutative, products such as:
,
There are, of, course, such projections, they form a partition of unity themselves, and thus we can build a measurement that will identify a random permutation
and leave it equal to some
after measurement. This is the essence of classical… all we have to do is enumerate
and measure using:
.
A quantum permutation meanwhile is impossible to pin down in such a way. As an example, consider the Kac-Paljutkin quantum group of order eight which can be represented as . Take
. Then
.
If you think for a moment this cannot happen classically, and the issue is that we cannot know simultaneously if and
… and if we cannot know this simultaneously we cannot pin down
to a single element of
.
No Quantum Cyclic Group
Suppose that is a quantum permutation (in
). We can measure where the quantum permutation sends, say, one to. We simply form the self-adjoint element:
.
The measurement will produce some … but if
is supposed to represent some “quantum cyclic permutation” then we already know the values of
from
, and so, after measurement,
,
.
The significance of the intersection is that whatever representation of we have, we find these subspaces to be
-invariant, and can be taken to be one-dimensional.
I believe this explains why there is no quantum cyclic group.
Question 1
Can we use a similar argument to show that there is no quantum version of any abelian group? Perhaps using together with the structure theorem for finite abelian groups?
No Quantum 
Let be represented as bounded operators on a Hilbert space
. Let
. Consider the random variable
.
Assume without loss of generality that then measuring
with
gives
with probability
, and the quantum permutation projects to:
.
Now consider (for any , using the fact that
and the rows and columns of
are partitions of unity:
(*)
Now suppose, again without loss of generality, that measurement of with
produces
, then we have projection to
. Now let us find the Birkhoff slice of this. First of all, as
has just been observed it looks like:
In light of (*), let us find . First let us normalise correctly to
So
Now use (*):
,
and as maps to doubly stochastic matrices we find that
is equal to the permutation matrix
.
Not convincing? Fair enough, here is proper proof inspired by the above:
Let us show . Fix a Hilbert space representation
and let
.
The basic idea of the proof is, as above, to realise that once a quantum permutation is observed sending, say,
, the fates of
and
are entangled: if you see
you know that
.
This is the conceptional side of the proof.
Consider which is equal to both:
.
This is the manifestation of, if you know , then two and one are entangled. Similarly we can show that
and
.
Now write
.
Similarly,
Which is equal to , that is
and
commute.
Question 2
Is it true that if every quantum permutation in a can be fully described using some combination of
-measurements, then the quantum permutation group is classical? I believe this to be true.
Quantum Alternating Group
Freslon, Teyssier, and Wang state that there is no quantum alternating group. Can we use the ideas from above to explain why this is so? Perhaps for .
A possible plan of attack is to use the number of fixed points, , and perhaps show that
commutes with
. If you know these two simultaneously you nearly know the permutation. Just for completeness let us do this with
:
tr(u)\x(1) | 1 | 2 | 3 | 4 |
0 | – | (12)(34) | (13)(24) | (14)(23) |
1 | (234),(243) | (134),(143) | (124),(142) | (123),(132) |
4 | e | – | – | – |
The problem is that we cannot assume that that the spectrum of is
, and, euh, the obvious fact that it doesn’t actually work.
What is more promising is
x(1)\x(2) | 1 | 2 | 3 | 4 |
1 | – | e | (234) | (243) |
2 | (12)(34) | – | (123) | (124) |
3 | (132) | (134) | – | (13)(24) |
4 | (142) | (143) | (14)(23) | – |
However while the spectrums of x(1) and x(2) are cool (both in ), they do not commute.
Question 3
Are there some measurements that can identify an element of and via a positive answer to Question 3 explain why there is no quantum
? Can this be generalised to
.
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.
In May 2017, shortly after completing my PhD and giving a talk on it at a conference in Seoul, I wrote a post describing the outlook for my research.
I can go through that post paragraph-by-paragraph and thankfully most of the issues have been ironed out. In May 2018 I visited Adam Skalski at IMPAN and on that visit I developed a new example (4.2) of a random walk (with trivial -dependence) on the Sekine quantum groups
with upper and lower bounds sharp enough to prove the non-existence of the cutoff phenomenon. The question of developing a walk on
showing cutoff… I now think this is unlikely considering the study of Isabelle Baraquin and my intuitions about the ‘growth’ of
(perhaps if cutoff doesn’t arise in somewhat ‘natural’ examples best not try and force the issue?). With the help of Amaury Freslon, I was able to improve to presentation of the walk (Ex 4.1) on the dual quantum group
. With the help of others, it was seen that the quantum total variation distance is equal to the projection distance (Prop. 2.1). Thankfully I have recently proved the Ergodic Theorem for Random Walks on Finite Quantum Groups. This did involve a study of subgroups (and quasi-subgroups) of quantum groups but normal subgroups of quantum groups did not play so much of a role as I expected. Amaury Freslon extended the upper bound lemma to compact Kac algebras. Finally I put the PhD on the arXiv and also wrote a paper based on it.
Many of these questions, other questions in the PhD, as well as other questions that arose around the time I visited Seoul (e.g. what about random transpositions in ?) were answered by Amaury Freslon in this paper. Following an email conversation with Amaury, and some communication with Uwe Franz, I was able to write another post outlining the state of play.
This put some of the problems I had been considering into the categories of Solved, to be Improved, More Questions, and Further Work. Most of these have now been addressed. That February 2018 post gave some direction, led me to visit Adam, and I got my first paper published.
After that paper, my interest turned to the problem of the Ergodic Theorem, and in May I visited Uwe in Besancon, where I gave a talk outlining some problems that I wanted to solve. The main focus was on proving this Ergodic Theorem for Finite Quantum Groups, and thankfully that has been achieved.
What I am currently doing is learning my compact quantum groups. This work is progressing (albeit slowly), and the focus is on delivering a series of classes on the topic to the functional analysts in the UCC School of Mathematical Sciences. The best way to learn, of course, is to teach. This of course isn’t new, so here I list some problems I might look at in short to medium term. Some of the following require me to know my compact quantum groups, and even non-Kac quantum groups, so this study is not at all futile in terms of furthering my own study.
I don’t really know where to start. Perhaps I should focus on learning my compact quantum groups for a number of months before tackling these in this order?
- My proof of the Ergodic Theorem leans heavily on the finiteness assumption but a lot of the stuff in that paper (and there are many partial results in that paper also) should be true in the compact case too. How much of the proof/results carry into the compact case? A full Ergodic Theorem for Random Walks on Compact Quantum Groups is probably quite far away at this point, but perhaps partial results under assumptions such as (co?)amenability might be possible. OR try and prove ergodic theorems for specific compact quantum groups.
- Look at random walks on quantum homogeneous spaces, possibly using Gelfand Pair theory. Start in finite and move into Kac?
- Following Urban, study convolution factorisations of the Haar state.
- Examples of non-central random walks on compact groups.
- Extending the Upper Bound Lemma to the non-Kac case. As I speak, this is beyond what I am capable of. This also requires work on the projection and quantum total variation distances (i.e. show they are equal in this larger category)
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.
Finally cracked this egg.
Preprint here.
I thought I had a bit of a breakthrough. So, consider the algebra of a functions on the dual (quantum) group . Consider the projection:
.
Define by:
.
Note
.
Note so
is a partition of unity.
I know that corresponds to a quasi-subgroup but not a quantum subgroup because
is not normal.
This was supposed to say that the result I proved a few days ago that (in context), that corresponded to a quasi-subgroup, was as far as we could go.
For , note
,
is a projection, in fact a group like projection, in .
Alas note:
That is the group like projection associated to is subharmonic. This should imply that nearby there exists a projection
such that
for all
… also
is subharmonic.
This really should be enough and I should be looking perhaps at the standard representation, or the permutation representation, or … but I want to find the projection…
Indeed …and
.
The punchline… the result of Fagnola and Pellicer holds when the random walk is is irreducible. This walk is not… back to the drawing board.
I have constructed the following example. The question will be does it have periodicity.
Where is the permutation representation,
, and
,
is given by:
.
This has (duh),
, and otherwise
.
The above is still a cyclic partition of unity… but is the walk irreducible?
The easiest way might be to look for a subharmonic . This is way easier… with
it is easy to construct non-trivial subharmonics… not with this
. It is straightforward to show there are no non-trivial subharmonics and so
is irreducible, periodic, but
is not a quantum subgroup.
It also means, in conjunction with work I’ve done already, that I have my result:
Definition Let be a finite quantum group. A state
is concentrated on a cyclic coset of a proper quasi-subgroup if there exists a pair of projections,
, such that
,
is a group-like projection,
and there exists
(
) such that
.
(Finally) The Ergodic Theorem for Random Walks on Finite Quantum Groups
A random walk on a finite quantum group is ergodic if and only if the driving probability is not concentrated on a proper quasi-subgroup, nor on a cyclic coset of a proper quasi-subgroup.
The end of the previous Research Log suggested a way towards showing that can be associated to an idempotent state
. Over night I thought of another way.
Using the Pierce decomposition with respect to (where
),
.
The corner is a hereditary
-subalgebra of
. This implies that if
and for
,
.
Let . We know from Fagnola and Pellicer that
and
.
By assumption in the background here we have an irreducible and periodic random walk driven by . This means that for all projections
, there exists
such that
.
Define:
.
Define:
.
The claim is that the support of ,
is equal to
.
We probably need to write down that:
.
Consider for any
. Note
that is each is supported on
. This means furthermore that
.
Suppose that the support . A question arises… is
? This follows from the fact that
and
is hereditary.
Consider a projection . We know that there exists a
such that
.
This implies that , say
(note
):
By assumption . Consider
.
For this to equal one every must equal one but
.
Therefore is the support of
.
Let . We have shown above that
for all
. This is an idempotent state such that
is its support (a similar argument to above shows this). Therefore
is a group like projection and so we denote it by
and
!
Today, for finite quantum groups, I want to explore some properties of the relationship between a state , its density
(
), and the support of
,
.
I also want to learn about the interaction between these object, the stochastic operator
,
and the result
,
where is defined as (where
by
).
.
An obvious thing to note is that
.
Also, because
That doesn’t say much. We are possibly hoping to say that .
Recent Comments