## Background

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:

#### Ergodic Theorem for Random Walks on 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:

#### Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group driven by a state is ergodic if and only if “X”.

## Irreducibility

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.

## Aperiodicity

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:

- if is/is not supported on the coset of a proper normal subgroup, then the random walk is periodic/aperiodic.
- if is/is not supported on the coset of a proper normal
*quasi*-subgroup, then the random walk is periodic/aperiodic.

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?

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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