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

In this section we want to represent the algebras on Hilbert spaces and obtain spatial implementations for the random walk. On a finite quantum group we can introduce an inner product

,

where and and is the Haar state. Because the Haar state is faithful we can think of as a finite dimensional Hilbert space. Further we denote by the norm associated to this inner product. We consider the linear operator

, .

It turns out that this operator contains all information about the quantum group and thus it is called its *fundamental operator*. We discuss some of its properties.

#### (a)

* is unitary.*

*Proof *: Using it follows that

.

A similar computation works for instead of . Thus is isometric and, because is finite dimensional, also unitary

I am confused here — I can’t see the second, third nor the last two equalities.

It can easily be checked using Sweedler’s notation that with the antipode the inverse can be written explicitly as

.

#### (b)

* satisfies the Pentagon Equation*

.

This is an equation on and we have used the *leg notation* , , .

*Proof *: Just a straight, fun, calculation

#### Remark

The pentagon equation expresses the coassociativity of the comultiplication . Unitaries satisfying the pentagon equation have been called *multiplicative unitaries*.

The operator of left multiplication by :

,

will often be written as in the following. It is always clear whether or is meant. We can also look at left multiplications as a faithful representation of the C*-algebra acting on itself. In this sense we have

#### (c)

for all .

*Proof *: Here and are left multiplication operators on . The formula can via another calculation

By left multiplication we can also represent a random walk on a finite quantum group . Then becomes an operator on a -fold tensor product of . To get used to it let us show how the pentagon equation is related to Proposition 3.1.

## Theorem 5.2

.

,

*where means restriction to and this left position gets the number zero.*

I can’t make head nor tail of this notation. The only thing that makes sense to me is something like

and acts something like:

.

Not much point doing the proof when I can’t understand the notation (the proof doesn’t really help with the notation either).

It is an immediate but remarkable consequence of this representation that we have a canonical way of extending our random walk to , the C*-algebra of all (bounded) linear operators on . Namely, we can for define the random variables

,

…

There is not much point going any further until I can clear up the notation of Theorem 5.2.

$latex $

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