Recall that for an ordinary group , its subgroup is a subset which is closed under the group operation. However, for abstract groups, the notion of a subset may not always be well defined. In that case, we say that is a subgroup of if there is some injective group homomorphism embedding . Specifying the form of the embedding is then an important datum. In contrast, in the case of matrix groups, we usually use the former strict definition – the embedding ought to be the identity. Similar distinction occurs for quantum groups.
Let be compact quantum groups. We say that is a quantum subgroup of if there is a surjective -homomorphism satisfying . The surjective -homomorphism then dualizes the embedding .
In the matrix case, we usually compare the quantum groups with respect to their fundamental representations. That is, suppose that and are compact matrix quantum group and and their respective fundamental representations. We say that is a quantum subgroup of , denoted , if the map extends to a surjective -homomorphism .
Suppose are compact matrix quantum groups with unitary fundamental representations. Then the following are equivalent.
Let be a Hopf -algebra. A set is called a coideal if
A coideal that is also a (-)ideal is called a *-biideal. A Hopf (*-)ideal is a (-)biideal that is invariant under the antipode, that is, . For any Hopf -algebra and a Hopf -ideal the quotient~ becomes a~Hopf -algebra with respect to the operations induced from .
Given compact quantum groups , the corresponding surjection defines a Hopf -ideal and . Conversely, given a compact quantum group , any Hopf -ideal defines a compact quantum subgroup such that .
Let be a compact quantum group and denote by its discrete dual. Suppose that is a Hopf -subalgebra. Then actually defines a compact quantum group and a discrete quantum group . We say that is quotient quantum group of and that is a discrete quantum subgroup of .
Let be a~compact quantum group and its quotient. Then is a full subcategory of . More concretely, consists of all representations such that .
In particular, we have . More precisely,