Explicit finite-group quantum-double data

Let $G$ be a finite group with identity $e$. Let ${\mathbb{C}}^G$ be the complex vector space of functions on $G$, with point-mass basis $\{{\delta }_h:h\in G\}$, and let $\mathbb{C}[G]$ be the group algebra with basis $\{g:g\in G\}$ and multiplication inherited from $G$. Define the $\mathrm{\mid }G{\mathrm{\mid }}^2$-dimensional vector space

$D(G):={\mathbb{C}}^G{\otimes }_{\mathbb{C}}\mathbb{C}[G],$

with basis $d_{h,g}:={\delta }_h\otimes g$. On basis elements prescribe the bilinear multiplication

$d_{h,g}{d}_{k,l}:={\mathbf{1}}_{h=gk{g}^{-1}}{d}_{h,gl},$

the linear coproduct and counit

$\mathrm{\Delta }(d_{h,g}):={\sum }_{ab=h}{d}_{a,g}\otimes d_{b,g},\epsilon (d_{h,g}):={\mathbf{1}}_{h=e},$

the unit vector and linear map

$1_D:={\sum }_{h\in G}{d}_{h,e},\eta (1):=1_D,$

the linear map $S(d_{h,g}):=d_{g^{-1}{h}^{-1}g,g^{-1}}$, and the conjugate-linear map

$d_{h,g}^{\ast }:=d_{g^{-1}hg,g^{-1}}\mathrm{.}$

Every displayed operation extends uniquely by the indicated linearity from the finite basis.

Here a Hopf algebra over $\mathbb{C}$ is a complex vector space $A$ with linear maps $m:A\otimes A\to A$, $\eta :\mathbb{C}\to A$, $\mathrm{\Delta }:A\to A\otimes A$, $\epsilon :A\to \mathbb{C}$, and $S:A\to A$. Writing $ab=m(a\otimes b)$ and $1_A=\eta (1)$, require $(ab)c=a(bc)$, $1_{A}a=a=a{1}_A$, $(\mathrm{\Delta }\otimes \mathrm{i}\mathrm{d})\mathrm{\Delta }=(\mathrm{i}\mathrm{d}\otimes \mathrm{\Delta })\mathrm{\Delta }$, and $(\epsilon \otimes \mathrm{i}\mathrm{d})\mathrm{\Delta }=\mathrm{i}\mathrm{d}=(\mathrm{i}\mathrm{d}\otimes \epsilon )\mathrm{\Delta }$ under the canonical tensor identifications. Require also $\mathrm{\Delta }(ab)=\mathrm{\Delta }(a)\mathrm{\Delta }(b)$, $\mathrm{\Delta }(1_A)=1_A\otimes 1_A$, $\epsilon (ab)=\epsilon (a)\epsilon (b)$, $\epsilon (1_A)=1$, and $m(S\otimes \mathrm{i}\mathrm{d})\mathrm{\Delta }=\eta \epsilon =m(\mathrm{i}\mathrm{d}\otimes S)\mathrm{\Delta }$, where $(a\otimes b)(c\otimes d)=ac\otimes bd$. A Hopf $\ast$-algebra means such a Hopf algebra equipped with a conjugate-linear involution $\ast$ that reverses multiplication, fixes the unit, and satisfies $\mathrm{\Delta }(a^{\ast })=\mathrm{\Delta }(a{)}^{\ast }$ and $\epsilon (a^{\ast })=\overline{\epsilon (a)}$ for every $a$, where $(x\otimes y{)}^{\ast }=x^{\ast }\otimes y^{\ast }$ on $D(G)\otimes D(G)$.

The group basis of $\mathbb{C}[G]$ is also taken orthonormal when it is used as a finite-dimensional edge Hilbert space. For a finite edge set $E$, choose a bijection to a finite region $\mathrm{\Lambda }\subset \mathbb{Z}$ and an enumeration of $G$ by $\{1,\dots ,q\}$, where $q=\mathrm{\mid }G\mathrm{\mid }\ge 1$. Then ${⨂}_{e\in E}\mathbb{C}[G]$ is the $q$-spin space ${\mathcal{H}}_{\mathrm{\Lambda }}$ of the finite local matrix algebra definition, and its operator algebra is $M_{q^{\mathrm{\mid }E\mathrm{\mid }}}(\mathbb{C})$. Different finite enumerations change only the basis labels.

Local spin matrix algebras and their factor inclusions

A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” §4

Mark de Wild Propitius and F. Alexander Bais, Discrete Gauge Theories, §2.1

Alexander Cowtan and Shahn Majid, “Quantum Double Aspects of Surface Code Models,” §3