Tensor product of quantum groups

Tensor product of quantum groups was defined by Wang in [Wan95] generalizing the group direct product.

Definition

Motivation: direct product of groups

Let $G$ and $H$ be groups. Then we can construct their direct product

$G\times H=\{(g,h)\mid g\in G,h\in H\}$

with group operation $(g_1,h_1)(g_2,h_2)=(g_1g_2,h_1h_2)$ . We can then identify the group $G$ with a subgroup $\{(g,e_H)\}_{g\in G}\subset G\times H$ and similarly $H\simeq(e_G,H)\subset G\times H$ . Then we can say that the elements of $G$ commute with the elements of $H$ in $G\times H$ in the sense that $gh=(g,e)(e,h)=(g,h)=(e,h)(g,e)=hg$ . In addition, the groups $G$ and $H$ can also be obtained as quotient groups of $G\times H$ .

If $G$ and $H$ are compact matrix groups with fundamental representations $u$ and $v$ , then $G\times H$ is also a compact group that can be represented by the direct sum $u\oplus v$ . For the associated C*-algebra of continuous functions, we have $C(G\times H)=C(G)\otimes C(H)$ as this holds for any compact spaces. We can also have a look on the comultiplication on $C(G\times H)$ . Take $f_1\in C(G)$ and $f_2\in C(H)$ and take $(g_1,h_1),(g_2,h_2)\in G\times H$ , then

$\eqalign{\Delta_{G\times H}(f_1\otimes f_2)((g_1,h_1),(g_2,h_2))&=(f_1\otimes f_2)((g_1g_2),(h_1h_2))\cr&=f_1(g_1g_2)f_2(h_1h_2)=\Delta_G(f_1)(g_1,g_2)\,\Delta_H(f_2)(h_1,h_2),}$

so $\Delta_{G\times H}(f_1\otimes f_2)=\Delta_G(f_1)\,\Delta_H(f_2)$ .

Nevertheless, the direct product is defined for any pair of groups, not only the compact ones. In particular, if $\Gamma_1$ and $\Gamma_2$ are discrete groups, then $\Gamma_1\times\Gamma_2$ defines a discrete group. For the associated group algebras, we obviously have $\C(\Gamma_1\times\Gamma_2)=\C\Gamma_1\odot\C\Gamma_2$ and hence $C^*(\Gamma_1\times\Gamma_2)=C^*(\Gamma_1)\otimes_{\rm max}C^*(\Gamma_2)$ .

Definition for compact quantum groups

Let $G$ and $H$ be compact quantum groups. We define [Wan95] their tensor product $G\times H$ to be the quantum group with underlying C*-algebra $C(G\times H):=C(G)\otimes_{\rm max}C(H)$ and comultiplication defined as

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

Formally, we should rather write $\Delta_{G\times H}(\iota_G(a)\otimes\iota_H(b))=(\iota_G\otimes\iota_G)(\Delta_G(a))\,(\iota_H\otimes\iota_H)(\Delta_H(b))$ , where $\iota_G\colon C(G)\to C(G)\otimes_{\rm max}C(H)$ and $\iota_H\colon C(H)\to C(G)\otimes_{\rm max}C(H)$ are the canonical inclusions.

The above mentioned construction indeed defines a compact quantum group $G\times H$ .

Taking two compact groups $G$ and $H$ , their quantum group tensor product coincides with the group direct product.

Considering an element $a\otimes b\in C(G)\otimes_{\rm max}C(H)$ , we usually omit the sign $\otimes$ . One way to view this is to consider $C(G)\otimes_{\rm max}C(H)$ as a quotient of $C(G)*_\C C(H)$ with respect to the relations $ab=ba$ for $a\in C(G)$ and $b\in C(H)$ . The other viewpoint is to consider $C(G)$ and $C(H)$ as subalgebras of $C(G)\otimes_{\rm max}C(H)$ . This also shows that $G$ and $H$ can be considered as quotient quantum groups of $G\times H$ . In addition, we also have that $C(G)$ and $C(H)$ are quotients of $C(G)\otimes_{\rm max}C(H)$ , so $G$ and $H$ are quantum subgroups of $G\times H$ .

Definition for compact matrix quantum groups

If $G$ and $H$ are compact matrix quantum groups, we can define the structure of a compact matrix quantum group on $G\times H$ .

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

$G\times H=(C(G)\otimes_{\rm max}C(H),u\oplus v)$

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

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 $C(G\times H)$ , 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 $G\times H$ is formed by $w^\alpha\otimes w^\beta$ with $\alpha\in\Irr G$ , $\beta\in\Irr H$ [Wan95].

Haar state

Let $G$ and $H$ be compact quantum groups and $h_G$ , $h_H$ the corresponding Haar states. Then the Haar state of $G\times H$ is of the form $h_G\otimes h_H$ [Wan95].