**The following is taken (almost) directly from the first draft of my PhD thesis.**

## The Quantisation Functor

This functor can be used to motivate the correct notion of (the algebra of functions on) a *quantum group.* Note that the ‘quantised’ objects that are arrived at via this ‘categorical quantisation’ are nothing but the established definitions so this section should be considered as little more than a motivation. The author feels that introductory texts on quantum groups could include these ideas and that is why they are included here. This quantisation is the translation of statements about a finite group, into statements about the algebra of functions on , .

This notion of quantisation sits naturally in category theory where two functors — the functor and the dual functor — lead towards a satisfactory quantisation.

### The Functor

The category of finite sets, , has the class of all finite sets as objects and functions for morphisms. Also of interest is the category of finite dimensional complex vector spaces, with linear maps for morphisms. There is a map, the map, , that associates to each object , an object — the complex vector space with basis . This map associates to each morphism a morphism , and it isn’t difficult to see that it is a covariant functor.

If and are finite sets then is also a finite set. This object is sent to by the functor. The following explains how to deal with , as well as presenting a number of other useful isomorphisms of vector spaces.

#### Theorem (Tensor Product Isomorphisms)

- Let and be finite sets. Then, under the isomorphism , .
- Let be a finite dimensional complex vector space. Then, under the isomorphism , .
- Let and be finite dimensional complex vector spaces. Then .

*Proof: *Standard results

Therefore, a morphism is sent to the linear map :

The dual map, , is a morphism in the category of finite dimensional vector spaces that sends a vector space to its dual and a linear map to its transpose:

It can be shown that for and that

With this result, and the fact that is linear, the dual functor is a contravariant endofunctor.

Call the composition of these two functors by the quantisation functor:

It will be seen that the image of a group under this functor is the algebra of functions on the group. This gives us a routine to quantise groups and related objects: apply the functor to objects, morphism and commutative diagrams in the category of finite sets to get quantised objects, morphisms and commutative diagrams in the category of finite dimensional vector spaces.

It will be seen that the image of a finite group under this functor, , has the structure of a*Hopf*-algebra: whose axioms are found simply by quantising the group axioms on . That is satisfies:

There are, however, vector spaces together with morphisms that also satisfy these axioms but aren’t the algebra of functions on any group — because the multiplication is no longer commutative. These are the algebras of functions on *quantum groups*:

”Algebras of functions” on quantum groups are algebras, , that satisfy the quantisations of the group axioms, except letting go of commutativity in the algebra means that is a virtual object.

## Algebra of Functions on a Group and the Group Ring

As a group is a ‘space’, it is natural to study the algebra of functions on it. Let be a finite group and let be the set of complex-valued functions on . There is a natural -algebra structure on defined by:

The unit is the indicator function . As in the previous discussion, there are relations that will always hold ‘up’ in as quantised versions of the relations ‘down’ in . The quantisation functor is used to see exactly what these relations look like in . Note that is referred to as the algebra of functions on and is a commutative -algebra.

Also associated to a finite group is another canonical algebra: the group ring. For a finite group, let be a complex vector space with basis elements . The scalar multiplication and vector addition are, for a and , the natural

and the multiplication is given by:

The vector space together with the multiplication is a complex associative algebra called the *group ring* of $G$. Take an element of :

If the elements of are considered complex-valued functions on via the embedding , , a quick calculation shows that this multiplication is nothing but the convolution:

The unit is . There is also an involution:

turning into a *-algebra. Note that is commutative if and only if is abelian. Considering as a Hilbert space with an orthonormal basis , acts on by left multiplication so can be seen as an algebra of linear operators on the Hilbert space and thus a -algebra with the operator norm.

Note that can be identified with the algebraic dual of via

and as is finite dimensional:

## Quantising Finite Groups

In this section the category theory approach is taken to the category of finite groups. A group is an object in together with morphisms , and that satisfy:

The second commutative diagram invokes the isomorphism while the third uses the maps , and .

Now apply the covariant functor to , the three morphisms and these three commutative diagrams. Firstly the image of is . The image of the group multiplication is the linear multiplication :

Note that and so :

Note that is the unit of and so denote by the *unit map*. The image of is the linear map , . Note

and denote . Finally

and denote .

The image of the commutative diagrams above are therefore given by:

Indeed, the first two commutative diagrams here show that together with and is an *algebra*.

To fully quantise the group, the contravariant dual functor must be applied to , the morphisms and the commutative diagrams. First note that . The multiplication has a dual:

where the last isomorphism can be seen as a consequence of . Embed the group in the group ring via and consider for :

This map , , is the *comultiplication* on . Note that is the indicator function on so that after the identification ,

Now consider the unit map , . The dual of this map is

Consider an element :

so that . This map is called the *counit*.

The inverse map has a dual which (via the embedding) is given by:

This map is called the *antipode*.

These are the most important dualisations of maps but there are two more namely and . Note that maps from to so that

Let and :

so that is just the pointwise multiplication on . Finally consider the map , . Its dual is the unit map of ( is the unit of the algebra ) as can be seen by taking any :

that is , i.e. .

Now that the morphisms have been identified:

applying the dual functor to the last set of commutative diagrams gives *coassociativity*, the *counital* property and the *antipodal* property:

The first two commutative diagrams here show that together with and is a *coalgebra*.

Now take an object in the category of finite vector spaces with morphisms that satisfy these ‘quantised’ axioms. Such an object is a finite Hopf-algebra and if it is non-commutative:

then it can be considered (if we make a few extra assumptions) the algebra of functions on a quantum group.

The preceding quantisation gives, more or less, the correct definition of a finite quantum group. To be more precise, we need to say quite a little more.

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