The manuals are available in the Copy Centre (at a cost of €14) and are required for Monday’s lecture.
The 15% Test 1 will be held in the Melbourne Hall, 16:00 Monday 14 October, Week 6. Ye all have lectures at 15:00 but I will request from your lecturers that ye be left go on time. There is a sample test in the notes.
If this is something you might be interested in, please download the Telegram app and set yourself up with an @ handle.
If you haven’t already, you are invited to take the following ‘Diagnostic Test’:
click here
This ‘Test’ does not go towards your grade, but allows me to give you some feedback on where you are in terms of material you have seen before that will be used in this module.
If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here.
We continued working with the dot product and then introduced the cross product.
We will look at the applications of vectors to work and moments. We might begin Chapter 2: Matrices.
Please feel free to ask me questions about the exercises via email or even better on this webpage.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>We had one lecture and after listening to me go on about the importance of mathematics to your programme we started the first chapter on Sets and Relations by looking at some number sets. We saw something new with the concept of the power set of a set. We also took the quick test.
In Week 2 we will start the first chapter proper. We will see something new with the concept of the power set of a set, and set identities, and we’ll explore Cartesian Products and perhaps introduce relations.
I am exploring the possibility of setting up a classroom group chat that would have the functionality of WhatsApp without having your phone number made public.
If this is something you might be interested in, please download the Telegram app and set yourself up with an @ handle.
We might talk about this again next week.
Tutorials start properly in Week 2.
I would urge anyone having any problems with material that isn’t being addressed in the tutorials to use the Academic Learning Centre. If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, you need to be aware of this resource. The timetable should be up some time next week.
You will get best results if you come to the helpers there with specific questions. Next week, some students will receive slips detailing areas of maths that they should brush up on.
There will be a 15% In-Class-Test in Week 6. Probably the Tuesday 14 October. Expect notice and a sample test about two weeks in advance.
Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>
The manuals are available in the Copy Centre. Please purchase ASAP. More information has been sent via email.
I am exploring the possibility of setting up a classroom group chat that would have the functionality of WhatsApp without having your phone number made public.
If this is something you might be interested in, please download the Telegram app and set yourself up with an @ handle.
We might talk about this again next week.
The Student Resources tab above contains some information about calculators.
Here is a list of some allowed and not allowed calculators.
If you have to purchase a calculator, my recommendation is that you purchase something like a Casio fx-83GT PLUS. This might be available in the CIT shop.
Tutorials, which are absolutely vital, start next week. There may be a split but this might not occur until Week 3.
In week one we had one and a half classes. One half class was given over to a general overview of MATH7019 and we spent about an hour introducing the topic of Curve Fitting including Lagrange Interpolation.
We will start talking about Least Squares curve fitting.
I would urge anyone having any problems with material that isn’t being addressed in the tutorials to use the Academic Learning Centre. If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, you need to be aware of this resource. The timetable should be up some time next week.
You will get best results if you come to the helpers there with specific questions. Next week, some students will receive slips detailing areas of maths that they should brush up on.
Assessment 1 will have a hand-in date the Friday of Week 5, 11 October. The Assignment is in the manual but I must also send on your personal data sets next week.
Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>,
exhibits a periodicity because
.
This shows that a necessary condition for ergodicity of a random walk on a finite classical group driven by is that the support of not be concentrated on the coset of a proper normal subgroup.
I had hoped that something similar might hold for the case of random walks on finite quantum groups but alas I think I have found a barrier.
Let be a proper normal subgroup so that is a finite group. Consider the pure state . In the classical case, the convolution of pure states remains pure. This is because for a finite group , the pure states are precisely the delta measures and the convolution of delta measures is a delta measure again:
.
In particular convolution powers of pure states remain pure.
The algebra is a subalgebra of consisting of functions constant on cosets. Suppose is concentrated on a coset . Then its support projection is less than the support projection of :
.
In the classical case, we have that:
.
Presumably this is a special case of, for states ,
.
What we are looking at here is which is not pure being comparable to which is pure. The state isn’t pure but it’s support is less than the support of a pure state on and so not ergodic.
An obvious candidate for a probability on a finite quantum group to be concentrated on the coset of a proper normal quantum subgroup … well what should be the analogue of for a quantum quotient group?
Something that comes to mind is that a pure state on would correspond to . Alternatively might be a minimal projection in the algebra of functions.
At any rate, we would want the convolution of pure states to remain pure. However this is not true in general in the quantum case.
Take the algebra of functions on a quantum group given by:
.
This has algebra:
.
Consider the pure state concentrated on the factor, say as simple as .
Then the pure state is no longer concentrated on a single summand but rather across the factors , , .
This means that is not pure.
While it may be the case that for any state such that
and ,
we can no longer say that is pure and so cannot know that .
Presumably it is possible to find a quantum group such that is a quotient of by a normal quantum subgroup so that
,
and a subspace of .
Now take an element of such that
.
Because is pure, classically we know that is not ergodic. We cannot be sure of this now in the quantum case.
We cannot argue that cannot be the whole of .
]]>
If you haven’t already, you are invited to take the following ‘Diagnostic Test’:
click here
This ‘Test’ does not go towards your grade, but allows me to give you some feedback on where you are in terms of material you have seen before that will be used in this module.
I am exploring the possibility of setting up a classroom group chat that would have the functionality of WhatsApp without having your phone number made public.
If this is something you might be interested in, please download the Telegram app and set yourself up with an @ handle.
We might talk about this again next week.
The manuals are available in the Copy Centre (at a cost of €14) and should be purchased as soon as possible.
Tutorial for BioEng2A: Wednesdays at 10:00 in B149
Tutorial for BioEng2B: Mondays at 17:00 in B189 (starts this Monday 16 September)
Tutorial for SET2: Mondays at 9:00 in B180 (starts Monday 23 September)
We began our study of Chapter 2, Vector Algebra. We looked at how to both visualise vectors and describe them algebraically. We learned how to find the magnitude and direction of a vector.
We will continue working with the vectors and hopefully learn how to add them and scalar multiply them, about displacement vectors, the vector product known as the dot product and perhaps then introduce the cross product.
The test will probably be the Monday of Week 5. Official notice will be given in Week 3. There is a sample test in the notes.
Please feel free to ask me questions about the exercises via email or even better on this webpage.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>Abstract: It is a folklore theorem that necessary and sufficient conditions for a random walk on a finite group to converge in distribution to the uniform distribution – “to random” – are that the driving probability is not concentrated on a proper subgroup nor the coset of a proper normal subgroup. This is the Ergodic Theorem for Random Walks on Finite Groups. In this talk we will outline the rarely written down proof, and explain why, for example, adjacent transpositions can never mix up a deck of cards. From here we will, in a very leisurely and natural fashion, introduce (and motivate the definition of) finite quantum groups, and random walks on them. We will see how the group algebra of a finite group is the algebra of functions on a finite quantum group. Freslon has very recently proved the Ergodic Theorem in this setting, and we present ongoing work towards an Ergodic Theorem in the more general finite quantum group setting; a result that would generalise both the folklore and group algebra Ergodic Theorems.
]]>Every finite quantum group has finite dimensional algebra of functions:
.
At least one of the factors must be one-dimensional to account for the counit , and if this factor is denoted , the counit is given by the dual element . There may be more and so reorder the index so that for , and for :
,
Denote by the states of . The pure states of arise as pure states on single factors.
In the case of the Kac-Paljutkin and Sekine quantum groups, the convolution powers of pure states exhibit a periodicity of sorts. Recall for these quantum groups that consists of a single matrix factor.
In these cases, for pure states of the form , that is supported on (and we can say a little more than is necessary), the convolution remains supported on because
.
If we have a pure state supported on , then because
,
then must be supported on, because of , .
Inductively all of the are supported on and the are supported on . This means that the convolutions powers of a pure state, in these cases, cannot converge to the Haar state.
The question is, do the results above about the image of and under the coproduct hold more generally? I believe that the paper of Kac and Paljutkin shows that this is the case whenever consists of a single factor… but does it hold more generally?
To find out we go back and do some sandboxing with the paper of Kac and Paljutkin. Which is a pleasure because that paper is beautiful. The blue stuff is my own scribbling.
Let be a finite quantum group with notation on the algebra of functions as above. Note that is commutative. Let
,
which is a central idempotent.
.
Proof: If , then for some , and , the mapping is a non-zero homomorphism from into commutative which is impossible.
If , then one of the , with ‘something’ in . Using the centrality and projectionality of , we can show that the given map is indeed a homomorphism.
It follows that , and so
Proof: Suppose that for some non-commutative . This means that there exists an index such that . Then for that factor,
is a non-null homomorphism from the non-commutative into the commutative.
We see that for all . Putting we get the result
The following says that is a group-like projection. We know from previous work that if a state is supported on a group-like projection that it will remain supported on it. In particular, any state supported on will remain there.
.
Proof: Since is a homomorphism, is an idempotent in . I do not understand nor require the rest of the proof.
is the algebra of functions on finite group with elements , and we write . The coproduct is given by .
We have:
,
,
,
as .
The element is a sum of four terms, lying in the subalgebras:
.
We already know what is going on with the first summand. Denote the second by . From the group-like-projection property, the last two summands are zero, so that
=\sum_{t\in G_1}\delta_{t}\otimes \delta_{t^{-1}i}+P_i$.
Since the are symmetric () mutually orthogonal idempotents, has similar properties:
for .
At this point Kac and Paljutkin restrict to , that is there is only one summand. Here we try to keep arbitrarily (finitely) many summands in .
Let the summand have matrix units , where . Kac and Paljutkin now do something which I think is a little dodgy, but basically that the integral over is equal on each of the , equal on each of the , and then zero off the diagonal.
It does follow from above that each is a projection.
Now I am stuck!
]]>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 .
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…
A WC*-subalgebra of is a W Hopf C*-algebra together with an injective morphism of WC*-algebras . Such a morphism is a C*-morphism such that
.
Note this is a different beast to the surjective C*-morphism I have previously seen.
Note that () and (with , and ) are trivial WC*-subalgebras of .
Note that in the finite classical case, the set of functions vanishing on a proper subgroup has the property that:
, and
That vanishes on implies that . Could ? Of course not — because then would not be closed under inverses. Similarly , and so for any . From above we know that if , that for all , either or is not an element of , that is either or vanishes on . This implies that
.
This makes a Hopf ideal.
Wang remarks that if the algebra of functions is finite, say , and a WC*-ideal, that and (but makes no claim to the above).
There is a slight error if the ideal is not proper. If the ideal is the whole of , then does not hold. Assume therefore that is a proper ideal. For , a linear map between finite vector spaces with kernel :
.
Let . As , either or . We know that , so that, using the antipodal property:
.
If , then , implying . Therefore we have that if is proper.
I am unable to show the stability of under , and have farmed this question out to MO (where it has been answered: using the below we have ).
The quotient of a WC*-algebra by a WC*-ideal has a unique WC*-algebra structure such that the quotient map is a morphism of WC*-algebras.
For every morphism of WC*-algebras. the kernel is a WC*-ideal. The image of is a WC*-algebra isomorphic to (as defined above). Furthermore this image is a WC*-subalgebra of .
Let be the morphism from above. If (should this be an (closed) ideal?), then there is a unique morphism of WC*-algebras , such that , where is the quotient map .
What does this look like classically?
1.Well first of all if the ideal is the full ideal then we are talking about the algebra of functions on the trivial group. So let us suppose that the ideal is proper. Ideals cannot contain invertible elements. This means that only functions with roots are in the ideal. Let be the set of roots of . Let be two elements in an ideal. Then has roots at . Note further that this cannot be non-empty, for it it is, then is invertible.
Therefore
,
is non-empty and is a subset of such that every element of vanishes on it.
Next question: is a group? The answer is yes. Let and . Using , and , we know that for all , , either or is in . Therefore
,
and either or is equal to zero. Therefore , and so is closed under multiplication. Is ?
Because is finite, every element has finite order, a least number such that . Therefore . We have shown that is closed under multiplication and so the result follows. Unfortunately this very “set of points” argument does not transfer easily to the quantum case.
Now presumably in this commutative case, we have
.
This follows from the map above.
2. To show this we probably have to show that a morphism of algebras on finite groups corresponds to a group homomorphism. I would suggest that every morphism is of the form for … I have spent some time now on this problem and perhaps it is a waste of time. I have shown if is the pullback of a group homomorphism, then
,
and I am certain this is a morphism of quantum groups. On the other hand I have shown that if is a morphism of classical , that with respect to the basis of delta functions, for all
and I am confident this will yield that is the pushback of a group homomorphism, but I am perhaps wasting my time. Let us move onto the proof of the quantum result.
Proof: 1. Let be the quotient . Define
,
where . This is well defined. For if , and
,
because , and it follows that is well defined.
Wang claims that is generated by .
2. Let . Define a C*-isomorphism by
,
that is . Under this isomorphism
identifies with .
Let us make a commutative diagram for all this:
Is a WC*-ideal? Well
.
We know that which implies , and so is a WC*-ideal.
It turns out that is an isomorphism of WC*-algebras from onto , and
,
and so is a WC*-subalgebra of under the natural injection
The following took me a little by surprise:
Let be a compact group. We have the correspondences:
, subgroups , WC*-ideals
with , WC*-subalgebras.
OK… a WC*subalgebra is an injective WC*-morphism. I am fairly sure that the map in question, in the classical case, is simply . I am reasonably confident that this is the case. Writing it down seems to be a little awkward.
A compact quantum is called a Wang subgroup of if there is a WC*-ideal of , , such that:
If there is a surjective morphism , is called embedded.
Let be Wang subgroup of a compact quantum group. This means there is a WC*-ideal such that . Let
be the quotient map.
is said to be normal if for every irreducible representation of , with matrix , the multiplicity of the trivial representation of ; , in the representation is either zero or the dimension of .
Hopefully someone here can help me with how this is a classical result.
Let be the right quotient space, defined by Podlés:
,
This is a compact quantum group, the right quotient of by . We also have a right quotient which is, in general, different to the right quotient. The full quantum group is normal in , and if the counit is bounded, so is the trivial group.
Let be a quantum subgroup of a quantum group with surjection , . is closed but this is not important at this moment.
What follows can also be done for right quotient spaces.
Define the left quotient
.
These are functions constant on left cosets of . Define
.
These are smooth functions constant on left cosets of . Define
.
This map is a projection of norm one (completely positive and completely bounded conditional expectation) from onto the the continuous functions on the right cosets.
Classically it maps a function on to a function on . The value that takes on is the average over .
We have that and this algebra is dense in .
Let be a subgroup of . The following are equivalent.
Before we tackle the proof, we must look at the Podlés results that the conditional expectation satisfies
(*)
If we can believe this we have for , as :
.
If we write down (*) as applied to the matrix element of an irreducible representation, , we find that it is actually rather trivial, and essentially due to the identities doing nothing on certain factors. In fact both are equal to:
.
Proof: (3) ($) In general
,
and similar for the left quotient. Letting , we have
.
Let be the multiplicity of the trivial representation of in . It is either or zero as shown (more or less) here.
I am going to leave the rest of the proof and move on
A quantum subgroup of a CQG is said to be normal if it satisfies the any of the equivalent conditions of the preceding proposition.
There is another condition given above about the multiplicity of the trivial representation… it is equivalent to condition (4).
The following seems a generalisation of the fact that every subgroup of an abelian group is normal.
Let . Let be a quantum subgroup of with surjection . Then is normal, where is the embedding of in , is discrete, and . Moreover where
]]>Let be the algebra of continuous functions on a compact matrix quantum group. Such an object is given by a matrix which generates as a C*-algebra. Furthermore, there exists a C*-algebra homomorphism such that
,
and both and are invertible in .
Any subgroup is such an object, with the given by . Furthermore
.
We say that is a representation if it is invertible and
.
The transpose is also invertible and so we have:
The C*algebra generated by the is also the algebra of continuous functions on a compact matrix quantum group.
Podlés proceeds to give some properties of representations of a compact matrix quantum group and introduces the index set . It is noted that the matrix element of the trivial representation is . Then the Haar state is presented as a functional such that
,
and for . Where is the unital *-algebra generated by the , the form a basis for . The Haar state is invariant in that for ,
which implies in particular that for all :
.
To measure the modular theory of the Haar state there exist matrices such that
.
It appears that Podlés now defines densities such that (dual element). It is important that this be done because not all functionals have densities.
We have that
,
and if we define
,
and (so that ), and
.
Perhaps this should merely be ? Well anyway, it’s like a column of .
Now Podlés wants to talk about subgroups and here we will make some notes to make sense of what Podlés does.
Consider the algebra of functions of continuous functions on a compact matrix quantum group , and assume that has fundamental representation .
We say that is a (compact) subgroup of if , and there exists a C*-homomorphism such that .
Podlés remarks that this must be a surjection. I suppose if is generated by the we can simply map the appropriate combination of to hit any combination of .
Let us try and reconcile this definition with the more standard definition. Where is the inclusion of into , is what I would call . The commutative diagram in the category of compact groups that says that is a subgroup of is given by:
.
The image of this commutative diagram under is the standard:
.
Suppose that . It is very easy to see that this satisfies the above.
Classically, in the finite picture, given a subgroup , and an element , one can see:
,
so it just chops bits off that are not relevant to outside . A simple example of what this looks like in the matrix case might be to take and
Then the are the coordinate functions but if either is two, and in this sense:
.
Getting back to Podlés.
Let be the algebra of continuous function on a compact quantum space and a compact quantum group. A C*-homomorphism is an action of on if
, and
.
I must admit I am not sure what the second condition is about. I assume it is something like the group acts transitively, I don’t know. The first condition, which doesn’t seem to include an analogue of , for actions of finite groups on finite sets at least, is the image under the functor of, where is a right action of on :
.
Let be a quantum space and a quantum group. Fix a C*-homomorphism . A subspace corresponds to a representation of if there exist basis elements , and
.
If is an action on then can be decomposed into subspaces corresponding to irreducible representations of .
Let be an action of a quantum group on a quantum space . Denote , .
For each , there exists a set and , such that
- corresponds to (. This is probably where the second condition comes in).
Subspaces corresponding to are contained in .
does not depend on the choice (?) of . Denoted by , called the multiplicity of in the spectrum of .
Proof: Set
.
Careful use of tensor product isomorphisms shows that that
.
Podlés claims that the densities of the elements of the dual basis to the generate . This is certainly true in the Kac case as the densities are equal to and these certainly span .
I have been struggling greatly with the second condition, the so called density condition. I understand that if one works in one has a counit and the extension of the condition gives this density condition, that it isn’t connected with a notion of transitivity. I am going to skip the rest of the proof because I am really interested in coset spaces and I might not need all this machinery to understand those… I can always go back. De Commer has a lot on actions that I can take a look at. I have a similar issue with a corollary to the theorem. Now however Podlés introduces the quotient spaces.
Let be a subgroup. Podlés identifies:
.
Let us show that if is classical, and a subgroup, that for a fixed , . Note firstly that
If is in , will leave it. Anything else will be killed by . So it will be left if there exists an such that
.
Therefore summing over is the same as summing over where . We have also . This gives:
It isn’t immediately clear but this does equal , and so . The condition is linear in therefore, classically,
,
that is functions constant on cosets . Note that the indicator functions are minimal projections in this subspace. I am not sure how much further structure we have… is it an algebra (in the quantum case)? I’m not sure. It is if
.
Now Podlés gives a completely bounded projection :
.
Classically this takes a function on and replaces it with a function constant on the cosets of . What values does it take on a coset ? The average of the function on
.
Podlés says that this projection has the property that
,
but I am not to sure of the relevance of this. Perhaps it allows the following make sense.
Define now a map
.
Podlés isn’t clear but I am fairly sure of what happens next. Take an irreducible representation of and map it to a representation of via . This representation is, apparently, not irreducible. This means that it can be decomposed into representations. The trivial representation, , appears times. Choose a basis such that these trivial representations appear ‘first’, in the top left hand corner, kind of (where represent representations):
,
where contains no non trivial representations.
It might be difficult to move to group-like-projections as they don’t have the representation theory as far as I know.
Well I think Podlés shows that only the trivial bits correspond to functions constant on cosets, so that, in this basis
and so the action is defined on these elements only
,
and essentially
.
Let us write out in detail what Podlés writes.
Let . If is a representation matrix for then so is .
Let us show this carefully. We are working with and so we will look at compatability and a counit condition. Firstly recall that . Apply this to to get
.
Using the counit condition in that
,
apply both to , together with the subgroup condition on to get:
,
so that indeed is a representation of .
Now Podlés decomposes into a direct sum of irreducible representations of , within which, as mentioned above, the trivial representation occurs with multiplicity . Now put all these trivial representations in the top left hand corner so that:
,
where is a direct sum of non-trivial representions of and there are copies of . Now what about
?
We know that of all the matrix elements of irreducible representations, except for . The first of these:
of these are the matrix elements of the trivial representation, and so we get one, but for , we get zero.
Consider now, with the same basis,
,
for . We find that for .
I think this is all I need from Podlés. Now onto Wang… and I will also want to talk about coideals at some stage.
]]>
I am trying to prove an Ergodic Theorem for Random Walks on Finite Quantum Groups. A random walk on a quantum group driven by is ergodic if the convolution powers converge to the Haar state .
The classical theorem for finite groups:
A random walk on a finite group driven by a probability is ergodic if and only if is not concentrated on a proper subgroup nor the coset of a proper normal subgroup.
Not concentrated on a proper subgroup gives irreducibility. A random walk is irreducible if for all , there exists such that .
Not concentrated on the coset of a proper normal subgroup gives aperiodicity. Something which should be equivalent to aperiodicity is if
is equal to one (perhaps via invariance ).
If is concentrated on the coset a proper normal subgroup , specifically on , then we have periodicity (), and , the order of .
In Markov chain theory, ergodicity is equivalent to irreduciblity and aperiodicity.
The theorem in the quantum case should look like:
A random walk on a finite quantum group driven by a state is ergodic if and only if “X”.
At the moment I have some part of X;the irreducibility bit. As is well known since Pal (1996), it is possible to have a probability not concentrated on a quantum subgroup be reducible. This led Franz & Skalski to generalise quantum subgroups to group-like-projections, which I will say correspond to quasi-subgroups following Kasprzak & Sołtan.
I have shown that if is concentrated on a proper quasi-subgroup , in the sense that for a group-like-projection , that so are the . The analogue of irreducible is that for all projections in , there exists such that . If is concentrated on a quasi-subgroup , then for all , , where .
I have also shown on the other hand that if the random walk is reducible that it must be concentrated on a proper quasi-subgroup. Franz & Skalski show that group-like-projections also have a correspondence with idempotent states. The Césaro Means
,
converge to an idempotent state . If for all then the also, so that (as the Haar state is faithful). I was able to prove that is supported on the quasi-subgroup given by the idempotent .
I believe this result — irreducible if and only if not concentrated on a quasi-subgroup — holds more generally than just in finite quantum groups.
Now onto aperiodicity. Below are propositions which may or may not be true. I am using quantum analogues which may or may not be appropriate. For example, if , then there is the sense of a quotient (which should also take into account a map that describes how sits inside . I am taking “concentrated on the coset of a normal subgroup” to be the same as, where is the support of , , where is a minimal projection in . I do not know is this appropriate.
The propositions to consider:
I do not even have a good definition for periodic… perhaps in the finite case the Haar element can be utilised. Perhaps we might say that the random walk is aperiodic if
.
I have quite a few sheets scrawled at this stage so I might record here some notes and possible approaches. Note that if we have a truly normal quasi-subgroup it may not make sense to reference .
– The smallest projection such that is well defined and called in the paper in preparation by the support projection of . Suppose that is a proper normal (quasi?-)subgroup. Let be a minimal projection in . We do not want to coincide with (a group-like-projection in the quasi case). Perhaps to test this we simply require ? As is positive, we can scale it to make it the density of a state, which I denote by . Perhaps the support of is a minimal projection in , different to .
Perhaps I should be using the stochastic operator more. Perhaps is a minimal projection if ?
– Perhaps even it might be possible to generate periodic behaviour on these , that if , which has support , this would make the convolution of powers of a cyclic group with identity . This would exhibit the periodicity inherent in the classical case and give a necessary condition for aperiodicity, which would be a nice partial result. Let us show a partial result in this direction. Let be constant the cosets of a normal quasi-subgroup given by a group-like-projection . This should mean, in particular, that
.
This means that . Now let be supported on a coset so that — where a minimal projection in . Write the above for :
.
Hit both sides with to get:
.
It should be the case, because is the support projection of , that the left hand side is equal to , that is remains supported on a coset. This mirrors
,
for cosets of a classical normal subgroup.
– To chase periodicity we can do other things. Perhaps the period might coincide with
.
This would be the time that the random walk would return to the coset . Perhaps in addition we might have
,
for . Perhaps the simpler might be a start.
– We would love to say something like the are all supported on minimal projections in . Might we have ?
– It may be true/useful that for the support projection, and the convolution in :
.
This might possibly help show that
,
which might possibly be a minimal projection in . This seems unlikely and difficult though.
– The theory should be illustratable by looking at the behavior in the Kac-Paljutkin quantum group. Moreover perhaps some periodicity from a truly quasi normal subgroup might be found…
– I am not sure is this useful/true, but in , it might be the case that the are pure states because the supports are minimal. It might be possible to get periodicity in that way.
– We could try and prove the classical result using the quantum machinery… and see can we write the sufficiency proof using the counit.
– At the end, I must see how this compares to the work of Amaury on , for a discrete group.
– This seems as good as place as any to record the work of Franz & Skalski; that a group like projection corresponds to an idempotent:
,
and an idempotent corresponds to a group-like-projection:
.
What now is that I do not know enough about normal subgroups and cosets to continue. I will now take a study of papers of Podlés and Wang.
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