## 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.

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