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Compact quantum group

Definition

A compact quantum group is a pair $G=(A,\Delta)$ of a unital C*-algebra $A$ and a unital $*$ -homomorphism

$\Delta\colon A\to A\otimes_{\rm min}A$

called co-multiplication which is co-associative, i.e.

$(\Delta\otimes\id)\circ\Delta=(\id\otimes\Delta)\circ\Delta,$

and satisfies the cancellation property, i.e. the spaces

$\Delta(A)(1\otimes A)=\span\{\Delta(a)(1\otimes b)\mid a,b\in A\},$

$\Delta(A)(A\otimes 1)=\span\{\Delta(a)(b\otimes 1)\mid a,b\in A\}$

are both dense in $A\otimes_{\rm min}A$ .

Examples coming from groups

Compact groups

Any compact group $G$ can be viewed as a compact quantum group. Indeed, put $A:=C(G)$ (the C*-algebra of continuous functions over $G$ ) and define $\Delta\colon C(G)\to C(G)\otimes C(G)\simeq C(G\times G)$ as

$(\Delta(f))(g,h):=f(gh),\qquad f\in C(G),\; g,h\in G.$

Then $(A,\Delta)$ forms a compact quantum group.

Conversely, we have the following. For any compact quantum group $(A,\Delta)$ such that $A$ is commutative, there exists a compact group $G$ such that $A\simeq C(G)$ and $\Delta$ is given as above.

This can be seen as a generalization/application of the Gelfand duality to the case of compact groups.

Discrete groups

Let $\Gamma$ be a discrete group. Put either $A:=C^*(\Gamma)$ (the group C*-algebra) or $A:=C^*_{\rm red}(\Gamma)$ (the reduced group C*-algebra). Define $\Delta(g):=g\otimes g$ . Then $\hat\Gamma:=(A,\Delta)$ is a compact quantum group.

This quantum group is called the dual of $\Gamma$ . Such a construction generalizes the Pontryagin duality. Indeed, if $\Gamma$ is abelian, then $C^*(\Gamma)$ is commutative and hence $\hat\Gamma$ is a compact group. It is the Pontryagin dual of $\Gamma$ .

Conversely, we also have the following. Let $G=(A,\Delta)$ be a compact quantum group satisfying $\tau\circ\Delta=\Delta$ (so-called cocommutativity of $A$ ), where $\tau\colon A\otimes A\to A\otimes A$ is the swapping isomorphism $x\otimes y\mapsto y\otimes x$ . Then there is a discrete group $\Gamma$ and a pair of unital surjective $*$ -homomorphisms

$C^*(\Gamma)\to A\to C^*_{\rm red}(\Gamma)$

intertwining the respective comultiplications.

Table of Contents

Compact quantum group

Definition

Examples coming from groups

Compact groups

Discrete groups

Various algebras associated to quantum groups

Further reading

References

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Table of Contents

Compact quantum group

Definition

Examples coming from groups

Compact groups

Discrete groups

Various algebras associated to quantum groups

Further reading

References

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