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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.
We denote
the span of matrix coefficients of all representations of . 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.
Conversely, for any Hopf -algebra with a positive integral , we can consider its universal C*-completion (see bellow) , 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 .