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## Quantum Subgroups

Let be a the algebra of functions on a finite or perhaps compact quantum group (with comultiplication ) and a state on . We say that a quantum group with algebra of function (with comultiplication ) is a quantum subgroup of if there exists a surjective unital *-homomorphism such that:

.

## The Classical Case

In the classical case, where the algebras of functions on and are commutative,

There is a natural embedding, in the classical case, if is open (always true for finite) (thanks UwF) of ,

,

with for , and otherwise.

Furthermore, is has the property that

,

which resembles .

In the case where is a probability on a classical group , supported on a subgroup , it is very easy to see that convolutions remain supported on . Indeed, is the distribution of the random variable

,

where the i.i.d. . Clearly and so is supported on .

We can also prove this using the language of the commutative algebra of functions on , . The state being supported on implies that

.

Consider now two probabilities on but supported on , say . As they are supported on we have

and .

Consider

,

that is is also supported on and inductively .

## Some Questions

Back to quantum groups with non-commutative algebras of functions.

- Can we embed in with a map and do we have , giving the projection-like quality to ?
- Is a suitable definition for being supported on the subgroup .

If this is the case, the above proof carries through to the quantum case.

- If there is no such embedding, what is the appropriate definition of a being supported on a quantum subgroup ?
- If does not have the property of , in this or another definition, is it still true that being supported on implies that is too?

## Edit

UwF has recommended that I look at this paper to improve my understanding of the concepts involved.

Slides of a talk given at the Irish Mathematical Society 2018 Meeting at University College Dublin, August 2018.

**Abstract** *Four generalisations are used to illustrate the topic. The generalisation from finite “classical” groups to finite quantum groups is motivated using the language of functors (“classical” in this context meaning that the algebra of functions on the group is commutative). The generalisation from random walks on finite “classical” groups to random walks on finite quantum groups is given, as is the generalisation of total variation distance to the quantum case. Finally, a central tool in the study of random walks on finite “classical” groups is the Upper Bound Lemma of Diaconis & Shahshahani, and a generalisation of this machinery is used to find convergence rates of random walks on finite quantum groups.*

## Distances between Probability Measures

Let be a finite quantum group and be the set of states on the -algebra .

The algebra has an invariant state , the dual space of .

Define a (bijective) map , by

,

for .

Then, where and , define the total variation distance between states by

.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative machinary shows that:

.

(supremum presentation)

In the classical case, using the test function , where , we have the probabilists’ preferred definition of total variation distance:

.

In the classical case the set of indicator functions on the subsets of the group exhaust the set of projections in , and therefore the classical total variation distance is equal to:

.

(Projection Distance)

In all cases the quantum total variation distance and the supremum presentation are equal. In the classical case they are equal also to the projection distance. Therefore, in the classical case, we are free to define the total variation distance by the projection distance.

## Quantum Projection Distance Quantum Variation Distance?

Perhaps, however, on truly quantum finite groups the projection distance could differ from the QTVD. In particular, a pair of states on a factor of might be different in QTVD vs in projection distance (this cannot occur in the classical case as all the factors are one dimensional).

Slides of a talk given at the Topological Quantum Groups and Harmonic Analysis workshop at Seoul National University, May 2017.

**Abstract** *A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis & Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. These ideas are generalised to the case of finite quantum groups.*

After a long time I have finally completed my PhD studies when I handed in my hardbound thesis (a copy of which you can see here).

It was a very long road but thankfully now the pressure is lifted and I can enjoy my study of quantum groups and random walks thereon for many years to come.

Let be a finite quantum group described by with an involutive antipode (I know this is true is the commutative or cocommutative case. I am not sure at this point how restrictive it is in general. The compact matrix quantum groups have this property so it isn’t a terrible restriction.) . Under the assumption of finiteness, there is a unique Haar state, on .

# Representation Theory

A *representation* of is a linear map that satisfies

The dimension of is given by . If has basis then we can define the *matrix elements* of by

One property of these that we will use it that .

Two representations and are said to be *equivalent*, , if there is an invertible *intertwiner* between them. An intertwiner between and is a map such that

We can show that every representation is equivalent to a unitary representation.

Timmermann shows that if is a maximal family of pairwise inequivalent irreducible representation that is a basis of . When we refer to “the matrix elements” we always refer to such a family. We define the span of as , the *space of matrix elements of* .

Given a representation , we define its *conjugate*, , where is the conjugate vector space of , by

so that the matrix elements of are .

Timmermann shows that the matrix elements have the following orthogonality relations:

- If and are inequivalent then for all and .
- If is such that the conjugate, , is equivalent to a unitary matrix (this is the case in the finite dimensional case), then we have

This second relation is more complicated without the assumption and refers to the entries and trace of an intertwiner from to the coreprepresention with matrix elements . If , then this intertwiner is simply the identity on and so the the entries and the trace is .

Denote by the set of unitary equivalence classes of irreducible unitary representations of . For each , let be a representative of the class where is the finite dimensional vector space on which acts.

# Diaconis-Van Daele Fourier Theory

In Group Representations in Probability and Statistics, Diaconis presents his celebrated Upper Bound Lemma for a random walk on a finite group driven by . It states that

,

where the sum is over all non-trivial irreducible representations of .

In this post, we begin this study by looking a the (co)-representations of a quantum group . The first thing to do is to write down a satisfactory definition of a representation of a group which we can quantise later. In the chapter on Diaconi-Fourier Theory here we defined a representation of a group as a group homomorphism

While this was perfectly adequate for when we are working with finite groups, it might not be as transparently quantisable. Instead we define a representation of a group as an action

.

such that the map , is linear.

Let be a group and let be the *C*-algebra *of the group . This is a C*-algebra whose elements are complex-valued functions on the group . We define operations on in the ordinary way save for multiplication

,

and the adjoint . Note that the above multiplication is the same as defining and extending via linearity. Thence is abelian if and only if is.

To give the structure of a quantum group we define the following linear maps:

, .

,

, .

The functional defined by is the Haar state on . It is very easy to write down the :

.

To do probability theory consider states on and form the product state:

.

Whenever is a state of such that implies that , then the distribution of the random variables converges to .

At the moment we will use the one-norm to measure the distance to stationary:

.

A quick calculation shows that:

.

When, for example, when are transpositions in , then we have

.

*Taken from Random Walks on Finite Quantum Groups by Franz & Gohm.*

## Theorem

*Let be a state on a finite quantum group . Then the Cesaro mean*

,

*converges to an idempotent state on , i.e. to a state such that .*

*Proof *: Let be an accumulation point of , this exists since the states on form a compact set. We have

.

I have no idea where the equality comes from.

Choose sequence such that , we get and similarly . By linearity this implies . If is another accumulation point of and a sequence such that , then we get and thus by symmetry (??). Therefore the sequence has a unique accumulation point, i.e. it converges

### Remark

If is faithful, then the Cesaro limit is the Haar state on (prove this).

### Remark

Due to *cyclicity *the sequence does not converge in general. Take, for example, the state (p.28) on the Kac-Paljutkin quantum group , then we have

,

but

.

*Taken from Random Walks on Finite Quantum Groups by Franz & Gohm*

In this section we will show how one can recover a classical Markov chain from a quantum Markov chain. We will apply a folklore theorem that says one gets a classical Markov process, if a quantum Markov process can be restricted to a commutative algebra.

We might like to do more. This result recovers a Markov chain — if the quantum process is in fact a random walk on a finite quantum group can we recover the group, the transition probabilities (yes), the driving probability measure??

## Conjecture

*If we restrict a random walk on a finite quantum group to a commutative subalgebra we can recover a random walk on a finite group.*

For random walks on quantum groups we have the following result.

## Theorem 6.1

*Let be a finite quantum group a random walk on a finite dimensional -comodule algebra , and a unital abelian sub-*-algebra of . The algebra is isomorphic to the algebra of functions on a finite set, say where .*

*If the transition operator of leaves invariant, then there exists a classical Markov chain with values in , whose probabilities can be computed as time-ordered moments of , i.e.*

*for all and .*

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