Just some notes on section 1 of this paper. Flags and notes are added but mistakes are mine alone.
Definition
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:
Proposition
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 .
Definition
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.
Definition
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 .
Theorem
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 grouplikeprojections 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 nontrivial 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.
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