A compact quantum group is a pair of a unital C*-algebra and a unital -homomorphism
called co-multiplication which is co-associative, i.e.
and satisfies the cancellation property, i.e. the spaces
are both dense in .
We usually denote .
Any compact group can be viewed as a compact quantum group. Indeed, put (the C*-algebra of continuous functions over ) and define as
Then forms a compact quantum group.
Conversely, we have the following. For any compact quantum group such that is commutative, there exists a compact group such that and is given as above.
This can be seen as a generalization/application of the Gelfand duality to the case of compact groups.
Let be a discrete group. Put either (the group C*-algebra) or (the reduced group C*-algebra). Define . Then is a compact quantum group.
This quantum group is called the dual of . Such a construction generalizes the Pontryagin duality. Indeed, if is abelian, then is commutative and hence is a compact group. It is the Pontryagin dual of .
Conversely, we also have the following. Let be a compact quantum group satisfying (so-called cocommutativity of ), where is the swapping isomorphism . Then there is a discrete group and a pair of unital surjective -homomorphisms
intertwining the respective comultiplications.
Let be a compact quantum group. There is a unique state on called the Haar state satisfying
This is a generalization of the Haar integral on a compact group.
A representation of a compact quantum group is a matrix with entries in satisfying
A representation is called non-degenerate if has a matrix inverse. It is called unitary if it is unitary as a matrix, i.e. .
There are several important statements generalizing the representation theory of compact groups
We denote by the set of classes of irreducible representations up to equivalence. For a given we denote by its representative, where is the corresponding matrix size.
We denote by the span of matrix coefficients of all representations of . Since every representation is a direct sum of irreducible ones, we can write
It holds that is a Hopf -algebra with respect to multiplication and comultiplication taken from , counit defined as , antipode defined as .
Moreover, is dense in , so it essentially contains all the information about the structure of the quantum group . Note however that there might exist several different C*-norms on and hence also several C*-completions of . As C*-algebras, those completions might be very different. Nevertheless, the quantum groups they describe are considered to be the same.
Let be the GNS representation of corresponding to the Haar state . We denote by the corresponding Hilbert space. It holds that the Haar state is faithful on (i.e. ). Hence, provides a faithful representation of on .
Conversely, for any Hopf -algebra with a positive integral , we can consider its universal C*-completion (see below) , which defines a compact quantum group. This provides an alternative algebraic definition of compact quantum groups.
Consider a compact quantum group . We may define the universal C*-norm on as
One needs to check that this is indeed a C*-norm. Then we denote by the completion of with respect to this norm. The C*-algebra then has the universal property that allows to extend the -homomorphism to . The pair then forms a compact quantum group called the universal or the full version of .
We denote by the closure of inside . Equivalently, it is the image of under the GNS-representation corresponding to the Haar state .
It can be checked that the comultiplication on extends to and hence is a compact quantum group called the reduced version of .
We denote by the weak closure of seen as a -subalgebra in . Such von Neumann algebras are the base object in the definition of a more general concept of a locally compact quantum group.
In the spirit of the Pontryagin duality, we can interpret any compact quantum group as a dual of some discrete quantum group , . We denote
We can make this idea more concrete by consider some kind of dual algebras that could be interpreted as algebras of functions (or rather sequences since is supposed to be discrete) over .
Let be a compact quantum group and denote by its discrete dual. We denote by the vector space dual of . This is a -algebra with respect to the following operations
where , . This algebra plays the role of the algebra of all functions (sequences) .
Given a (unitary) representation of , that is, a corepresentation of , we can define a (-)representation as .
Since with form a vector space basis, we have that any is determined by the numbers . Hence, we have
where the isomorphism is provided by .
Replacing the direct product by algebraic direct sum, we obtain an algebra denoted by corresponding to finitely supported sequences on . Taking the direct sum or direct sum, we can define also the algebras or . Using the direct sum, we arrive with the Banach space , which is the predual of .
The algebra is actually a Hopf -algebra with respect to the following operations
where , . Note that these operations can actually be defined also on , but the comultiplication would map with the inclusion being strict whenever is infinite dimensional.
Note also that the multiplication in is transformed into comultiplication on and the comultiplication on is transformed into multiplication on . In particular, is commutative, resp. cocommutative if and only if is cocommutative, resp. commutative.
A representation of the discrete dual on a Hilbert space is an element satisfying
where and is defined similarly adding the identity to the ``middle leg''. The equation hence essentially coincides with the equation defining representations of compact quantum groups. The only change is that here we formulate the definition also for infinite-dimensional representations.
Similarly as above, any (unitary) representation of induces a (-)representation of the algebra . Indeed, take any . We can decompose this element as a sum , where . Then, we can define as .
A compact quantum group is called finite if the associated C*-algebra is finite-dimensional. In this case, all other associated algebras coincide, so
The same hence holds for the dual algebras, which are also finite-dimensional
In particular, those algebras are unital C*-algebras and hence define a finite compact quantum group . A finite quantum version of the Pontryagin duality then says that .
In particular, any compact quantum group that is finite is also discrete (i.e. a dual of a compact one). In the formalism of locally compact quantum groups, one can formulate also the converse statement. If a locally compact quantum group is compact and discrete, then it is finite. Indeed, discreteness means that the associated reduced C*-algebra is of the form . The compactness then means that this C*-algebra is unital, which implies that the direct sum has to be finite.