In the case of a finite classical group , we can show that if we have i.i.d. random variables , that if , for a coset of a proper normal subgroup , that the random walk on driven by , the random variables:

,

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 .

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