Quantum subgroups

Definition of a quantum subgroup

Recall that for an ordinary group $G$ , its subgroup is a subset $H\subset G$ 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 $H$ is a subgroup of $G$ if there is some injective group homomorphism $\iota$ embedding $H\to G$ . 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 $H\to G$ ought to be the identity. Similar distinction occurs for quantum groups.

Compact quantum groups

Let $G,H$ be compact quantum groups. We say that $H$ is a quantum subgroup of $H$ if there is a surjective $*$ -homomorphism $\phi\colon\Pol G\to\Pol H$ satisfying $(\phi\otimes\phi)\circ\Delta_G=\Delta_H\circ\phi$ . The surjective $*$ -homomorphism $\phi$ then dualizes the embedding $H\to G$ .

Compact matrix quantum groups

In the matrix case, we usually compare the quantum groups with respect to their fundamental representations. That is, suppose that $G$ and $H$ are compact matrix quantum group and $u$ and $v$ their respective fundamental representations. We say that $H$ is a quantum subgroup of $G$ , denoted $H\subset G$ , if the map $u_{ij}\mapsto v_{ij}$ extends to a surjective $*$ -homomorphism $\Pol G\to\Pol H$ .

Equivalent definitions by relations and representation categories

Suppose $G,H$ are compact matrix quantum groups with unitary fundamental representations. Then the following are equivalent.

$H\subset G$ ,
$I_H\supset I_G$ ,
$\FundRep_H(w_1,w_2)\supset\FundRep_G(w_1,w_2)$ for all $w_1,w_2$ .

Constructing quantum subgroups

Hopf *-ideals

Let $A$ be a Hopf $*$ -algebra. A set $I\subset A$ is called a coideal if

$\Delta(I)\subset I\odot A+A\odot I\quad\hbox{and}\quad\epsilon(I)=0.$

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, $S(I)\subset I$ . For any Hopf $*$ -algebra $A$ and a Hopf $*$ -ideal $I\subset A$ the quotient~ $A/I$ becomes a~Hopf $*$ -algebra with respect to the operations induced from $A$ .

Ideals and quantum subgroups

Given compact quantum groups $H\subset G$ , the corresponding surjection $\phi\colon\Pol G\to\Pol H$ defines a Hopf $*$ -ideal $I=\ker\phi$ and $\Pol H\simeq\Pol G/I$ . Conversely, given a compact quantum group $G$ , any Hopf $*$ -ideal $I\subset\Pol G$ defines a compact quantum subgroup $H\subset G$ such that $\Pol H=\Pol G/I$ .

Quantum quotients and discrete quantum subgroups

Definition

Let $G$ be a compact quantum group and denote by $\Gamma:=\hat G$ its discrete dual. Suppose that $A_0\subset\Pol G$ is a Hopf $*$ -subalgebra. Then $A_0$ actually defines a compact quantum group $G_0$ and a discrete quantum group $\Gamma_0=\hat G_0$ . We say that $G_0$ is quotient quantum group of $G$ and that $\Gamma_0$ is a discrete quantum subgroup of $\Gamma$ .

Representations

Let $G$ be a~compact quantum group and $H$ its quotient. Then $\Rep_H$ is a full subcategory of $\Rep_G$ . More concretely, $\Rep_H$ consists of all representations $u\in\Rep_G$ such that $u_{ij}\in\Pol H\subset\Pol G$ .