论文标题
Kitaev量子双模型中边界的代数方面
Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model
论文作者
论文摘要
我们提供基于子组$ k \ subseteq g $的边界的系统处理，并在批量上使用Kitaev Quantum double $ d（g）$模型。边界站点是$*$ - subalgebra $ξ\ subseteq d（g）$的表示形式，我们将其结构阐述为强$*$ - quasi-hopf代数依赖于横向$ r $的选择。我们为$ d（g）$的不可约表示的分解公式拉回到$ξ$。我们还明确地提供了$ξ$ - 模块类别和$ g $的$ k $ -bimodules类别的单体等效性，并用它来证明$ r $的不同选择与Drinfeld Cochain Twists相关。示例包括$ s_ {n-1} \ subset s_n $，以及与$ξ$也是hopf Quasigroup的八元有关的示例。作为我们治疗的应用，我们将基于$ k = g $的边界的斑块水平和$ k = \ {e \} $垂直研究，并显示如何使用晶格手术技术在量子计算机中使用它们。
We provide a systematic treatment of boundaries based on subgroups $K\subseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Ξ\subseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra dependent on a choice of transversal $R$. We provide decomposition formulae for irreducible representations of $D(G)$ pulled back to $Ξ$. We also provide explicitly the monoidal equivalence of the category of $Ξ$-modules and the category of $G$-graded $K$-bimodules and use this to prove that different choices of $R$ are related by Drinfeld cochain twists. Examples include $S_{n-1}\subset S_n$ and an example related to the octonions where $Ξ$ is also a Hopf quasigroup. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=\{e\}$ vertically and show how these could be used in a quantum computer using the technique of lattice surgery.
