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.
]]>
From this paper I will look at:
Idempotent states on quantum groups correspond with “subgroup-like” objects. In this work, on LCQG, the correspondence is with quasi-subgroups (the work of Franz & Skalski the correspondence was with pre-subgroups and group-like projections).
Let us show the kind of thing I am trying to understand better.
Let be the algebra of function on a finite quantum group. Let be concentrated on a pre-subgroup . We can associate to a group like projection .
Let, and this is another thing I am trying to understand better, this support, the support of be ‘the smallest’ (?) projection such that . Denote this projection by . Define similarly. That are concentrated on is to say that and .
Define a map by
(or should this be or ?)
We can decompose, in the finite case, .
Claim: If is concentrated on , … I don’t have a proof but it should fall out of something like together with the decomposition of above. It may also require that is a trace, I don’t know. Something very similar in the preprint.
From here we can do the following. That is a group-like projection means that:
Hit both sides with to get:
.
By the fact that are supported on , the right-hand side equals one, and by the as-yet-unproven claim, we have
.
However this is the same as
,
in other words , that is remains supported on . As a corollary, a random walk driven by a probability concentrated on a pre-subgroup remains concentrated on .
Back to the paper, the “subgroup-like” objects are called quasi-subgroups. They are related to group-like projections.
I don’t fully have the functional analysis background (to… read this paper?!) to fully understand these objects, but we are denoting by the reduced -algebra of the locally compact quantum group . Let be the universal -algebra, and be the von Neumann algebra.
Denote the right Haar measure on by and the GNS Hilbert space of .
A von Neumann subalgebra is a left coideal if .
Let be a state on . The dual of this universal C*-algebra is Banach together with the convolution. Denote the set of idempotent states on the universal C*-algebra by .
Where is the half-lifted Kac-Takesaki Operator (??), the formula
,
for , defines a normal conditional expectation on and denote by its range:
,
which is a left-coideal in . Let
.
us a group-like projection in and is the orthogonal projection onto .
There is a bijective correspondence between:
.
Let . We say that dominates if (I would usually be doing something similar with the support, perhaps… perhaps not) and we write . This is equivalent to (which could be something to use in the finite/compact case). The convention is the opposite to that of Franz & Skalski so beware!
Let . TFAE
Let be a compact subgroup. This gives a surjective *-homomorphism from the universal C*-algebra of down to the universal C*-algebra of . We then have
.
This is a Haar type idempotent. Classically, all idempotents arise like this but not in the quantum case. In the quantum case let an idempotent correspond to a quasi-subgroup of . Under this line of thinking, the coideal corresponds to the von Neumann algebra of essentially bounded functions on the set of cosets of the quasi-subgroup.
Let . The von Neumann algebra generated by and , is an (integrable, -invariant) coideal, and thus of the form for some . Denote , and the corresponding quasi-subgroup as the intersection of the quasi-subgroups given by and . This operation is commutative and associative.
At this point I have no intuition for the following as I don’t understand what is… if we denote the above by , this vee is an associative and commutative map from pairs of idempotents to … hang on…
We call the quasi-subgroup corresponding to the quasi-subgroup generated by and . If , we say the quasi-subgroup is non-compact.
In particular, if is compact, then is non-zero, because is unital, and hence the set of states is closed in the weak* topology.
For , is the smallest (left) coideal containing both and . In particular, . Also is the largest idempotent dominated by both :
,
with a similar result for the quasi-group generated by .
Let such that
Then .
Note TFAE
Consider the reducing morphism
,
we embed the predual into (???). The predual is a closed ideal of the Banach algebra given by . Consider idempotents that lie in this ideal, called normal idempotents:
.
Such states correspond to open compact quasi-subgroups. The following generalises the fact that if a subgroup of a topological group contains an open subgroup, then it is itself open.
For any , , such that , .
Proof: Recall that the predual of the von Neumann algebra is an ideal in the convolution algebra. Therefore is in the predual also, and is thus an open compact quasi-subgroup
If is a discrete quantum group, then all quasi-subgroups are open compact.
Proof: Presumably is in the predual (something to do with (co)amenability?), and of course any has , and thus corresponds to an open compact quasi-subgroup
For each open compact quasi-subgroup, consider its left kernel
.
This left-kernel is a -weakly closed left ideal in the algebra of measurable functions, and hence of the form (just what I need?!) for a unique projection . A measurable is in this left kernel if and only if . Note as a projection, itself satisfies this and thus , and by Schwarz for any measurable (yes, this is very useful). Moreover putting , for all measurable we have:
.
If is a positive, measurable, and , then (yelp!). Indeed for some , so that for some measurable . Thus .
Now I think I understand what is — in fairness the paper spoke about it. Think classically, indeed finitely. Let be the uniform distribution on a subgroup . The big scary definition for above… I am fairly sure is nothing as bad in this easier category than
.
Let . Note
.
Let . Then
.
Therefore this conditional expectation takes functions defined on the group to functions defined on the left cosets of .
We want to say that is the set of functions constant on left cosets. This will feed into the following lemma.
Let be an open compact quasi-subgroup. Then
Let us illustrate the first (one direction) in the finite, classical case. Let be constant on left cosets of . That is
.
We have that is the projection while . We have
,
and
.
We can show that
.
Hitting both with shows that they are equal. Indeed both are equal to zero if . If then both are equal to .
Proof: Assume that . Hit both sides with . What we get (!) is the right-hand side gives
.
Now the left-hand side gives:
.
That is , that is is constant on left cosets.
Now assume that is constant on left cosets. Then so will (I actually can’t see this… or the next part)
…
Let be a projection, satisfying
,
, and . Then is group-like.
Recall that is a projection such that for functions such that , .
Let be an open compact quasi-subgroup. Then is a group-like projection.
Call the support projection of .
The support projection of any open compact quasi-subgroup is constant on conjugacy-classes (like one and zero). Moreover is minimal (?) and central among projections constant on conjugacy classes.
For any open compact quasi-subgroup, for
.
One more result of interest (to me).
Let give open compact quasi-subgroups. Then iff .
]]>Some of these problems have since been solved.
Consider a on a finite quantum group such that where
,
with . This has a positive density of trace one (with respect to the Haar state ), say
,
where is the Haar element.
An element in a direct sum is positive if and only if both elements are positive. The Haar element is positive and so . Assume that (if , then for all and we have trivial convergence)
Therefore let
be the density of .
Now we can explicitly write
.
This has stochastic operator
.
Let be an eigenvalue of of eigenvector . This yields
and thus
.
Therefore, as is also an eigenvector for , and is a stochastic operator (if is an eigenvector of eigenvalue , then , contradiction), we have
.
This means that the eigenvalues of lie in the ball and thus the only eigenvalue of magnitude one is , which has (left)-eigenvector the stationary distribution of , say .
If is symmetric/reversible in the sense that , then is self-adjoint and has a basis of (left)-eigenvectors and we have, if we write ,
,
which converges to (so that ).
If is not reversible, it is a standard argument to show that when put in Jordan normal form, that the powers converge and thus so do the
Uses Simeng Wang’s . Result holds for compact Kac if the state has a density.
is a minimal projection (coset) in the quotient space of the normal subgroup (to be double checked) given by
I have a proof that reducible is equivalent to supported on a pre-subgroup.
]]>If you do have time at the end of the exam, go through each of your answers and ask yourself:
On Monday, and Wednesday PM, we finished the module by looking at triple integrals.
The Wednesday 09:00 lecture was a tutorial along with most of Wednesday PM and the Thursday class.
Monday is a bank holiday.
In the Wednesday 09:00 lecture we will work on the Summer 2018 Paper and hopefully finish it before the end of the Thursday 10:00 lecture.
If we finish the Summer 2018 paper early (unlikely), any extra time will be dedicated to one-to-one help.
These are not always found in your programme selection — most of the time you will have to look here.
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..
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If you do have time at the end of the exam, go through each of your answers and ask yourself:
with comments, should be emailed to you this week. Test 2 Marking Scheme here.
We had some tutorial time from 18:00 – 19:00 before the test for those who could make it.
We then looked at sections 4.2 (completing the square) and 4.3 (work).
We did not have time to complete Example 4 on p.171, nor section 4.4.2 (centre of gravity of solid).
Both what we did in class and what we didn’t have time for can be found here.
Very Important Note: centres of gravity of solids do not appear on the summer exam paper. More information next week.
If you want to do some work on Chapter 4, Further Integration before starting more general revision you could look at:
There is an exam paper (Winter 2018) at the back of your notes — I will go through this on the board from 19:00-22:00 Tuesday night.
Past exam papers (MATH6040 runs in Semester 1 and Semester 2) may be found here.
Recall that this module is MATH6040: Technological Maths 201 and not MATH6015: Technological Maths 2.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>Has been corrected and results emailed to you.
Some remarks on common mistakes here.
We had a systems of differential equations tutorial Monday and before looking at double integrals.
We will look at triple integrals and then have one or two tutorials on. Possibly Wednesday 09:00 for double integrals and Thursday for triple integrals.
We will review the Summer 2018 paper.
Please feel free to ask me questions about the exercises via email or even better on this webpage.
These are not always found in your programme selection — most of the time you will have to look here.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..
]]>Test 2, worth 15% and based on Chapter 3, will now take place Week 12, 30 April. There is a sample test [to give an indication of length and layout only] in the notes (marking scheme) and the test will be based on Chapter 3 only.
More Q. 1s (on the test) can be found on p.112; more Q. 2s on p. 117; more Q. 3s on p.125 and p.172, Q.1; more Q. 4s on p.136, and more Q. 5s on p. 143.
Chapter 3 Summary p. 144.
Please feel free to ask me questions via email or even better on this webpage.
Once you are prepared for Test 2 you can start looking at Chapter 4:
We had some tutorial time from 18:00-19:00 for further differentiation (i.e. for Test 2, after Easter).
We completed our review of antidifferentiation before starting Chapter 4 proper.
We looked at Integration by Parts and centroids.
For those who could not make it here is some video and slides from what we did after the video died.
We will have some tutorial time from 18:00-19:00 for further differentiation.
We will have Test 2 from 19:00-20:05.
Then we will look at completing the square, centres of gravity, and work.
We will look the Winter 2018 paper at the back of your manual.
These are not always found in your programme selection — most of the time you will have to look here.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>