Free product of quantum groups

The following construction was defined by Wang in [Wan95]

Definition

Motivation for the compact case: Direct product of groups

Given two (compact) groups $G$ and $H$ , we can construct their direct product. Such a construction is generalized to the quantum case as the tensor product of quantum groups. Here, we have $C(G\times H)=C(G)\otimes C(H)$ . In particular, the two factors $C(G)$ and $C(H)$ commute.

If we start with some pair of free compact quantum groups (such as the free orthogonal quantum group $O_N^+$ ), we would like to generalize the direct product in a way, that the resulting C*-algebra will again be as free as possible. To achieve that, we replace the tensor product by the C*-algebraic free product.

Definition for compact quantum groups

Let $G$ and $H$ be compact quantum groups. We define [Wan95] their dual free product $G\hatstar H$ to be the quantum group with underlying C*-algebra $C(G\hatstar H):=C(G)*_\C C(H)$ and comultiplication the unique unital $*$ -homomorphism satisfying

$\Delta_{G\hatstar H}(a)=\Delta_G(a),\quad\Delta_{G\hatstar H}(b)=\Delta_H(b)\qquad\hbox{for all $a\in C(G)$, $b\in C(H)$.}$

Formally, we should rather write $\Delta_{G\hatstar H}(\iota_G(a))=(\iota_G\otimes\iota_G)(\Delta_G(a))$ , $\Delta_{G\hatstar H}(\iota_H(b))=(\iota_H\otimes\iota_H)(\Delta_H(b))$ , where $\iota_G$ and $\iota_H$ are the canonical inclusions into $C(G)*_\C C(H)$ .

Definition for compact matrix quantum groups

Let $G=(C(G),u)$ and $H=(C(H),v)$ be compact matrix quantum groups. Then

$G\hatstar H=(C(G)*_\C C(H),u\oplus v)$

is also a compact matrix quantum group. It is a matrix realization of the dual free product of $G$ and $H$ as defined above.

Motivation in the discrete case: Free product of groups

Given two discrete groups $\Gamma_1$ , $\Gamma_2$ , we can construct their free product $\Gamma_1*\Gamma_2$ . Note that the above definition of the dual free product should not be interpreted this way, because the freeness in case of the dual free product appears in the C*-algebra multiplication, not in the comultiplication. This is the reason for the word dual in the name. (And indeed, free product of the compact groups is a group that is never compact; in contrast, dual free product of two compact groups is a compact quantum group, but not a group).

However, dualizing the definition above, we may indeed generalize the definition of a free product to the case of discrete quantum groups.

Definition

Let $\Gamma_1$ and $\Gamma_2$ be discrete quantum groups. We define their free product to be the quantum group $\Gamma_1*\Gamma_2:=\widehat{\hat\Gamma_1\hatstar\hat\Gamma_2}$ .

In case when $\Gamma_1$ and $\Gamma_2$ are discrete groups, the above defined direct and free product exactly corresponds to the classical construction for groups.

Remark on the notation

The notation using the hat and calling the product dual free product is rather new. It was first used probably in [DFSW13]. Some authors omit the word dual and denote the product just by asterisk $*$ . In most cases, it should be clear, which product is meant – whether the dual free product or the free product – since the former is defined only in the compact case, whereas the latter only in the discrete case. It may, however, cause some confusion if one works with finite (quantum) groups.

Properties

Irreducible representations

Let $G$ and $H$ be compact quantum groups. Let $\{u^\alpha\}_{\alpha\in\Irr G}$ and $\{v^\beta\}_{\beta\in\Irr H}$ be complete sets of irreducible representation of $G$ and $H$ . Denote by $\iota_G$ and $\iota_H$ the embeddings of $C(G)$ and $C(H)$ into $C(G\hatstar H)$ , respectively, and denote $w^\alpha_{ij}:=\iota_G(u^\alpha_{ij})$ and $w^\beta_{ij}:=\iota_H(v^\beta_{ij})$ . Then a complete set of irreducible representations of $G\hatstar H$ is formed by the trivial representation together with

$w^{\gamma_1}\otimes w^{\gamma_2}\otimes\cdots\otimes w^{\gamma_n},$

where $\gamma_i\in\Irr G\cup\Irr H$ are non-trivial representations such that the sets $\Irr G$ and $\Irr H$ alternate, so if $\gamma_i\in\Irr G$ , then $\gamma_{i+1}\in\Irr H$ and vice versa. [Wan95]

Haar state

Let $G$ and $H$ be compact quantum groups and let $h_G$ , $h_H$ , respectively be the corresponding Haar states. Then the Haar state on $G*H$ is given by the free product $h=h_G*h_H$ . [Wan95]

References

[Wan95] Shuzhou Wang, 1995. Free products of compact quantum groups. Communications in Mathematical Physics, 167(3), pp.671–692.

[DFSW13] Matthew Daws, Pierre Fima, Adam Skalski, Stuart White, 2013. The Haagerup property for locally compact quantum groups. Journal für die reine und angewandte Mathematik, 2016(711), pp.189–229.

User Tools

Site Tools

Table of Contents

Free product of quantum groups

Definition

Motivation for the compact case: Direct product of groups

Definition for compact quantum groups

Definition for compact matrix quantum groups

Motivation in the discrete case: Free product of groups

Definition

Remark on the notation

Properties

Irreducible representations

Haar state

References

Page Tools