论文标题
基塔耶模型的泊松几何类似物
Poisson-geometric analogues of Kitaev models
论文作者
论文摘要
我们定义了Kitaev晶格模型的泊松几何类似物。它们是从嵌入式图$γ$上的Kitaev模型中获得的，它通过使用Poisson-Lie组G代替其HOPF代数数据。 每个边缘都被分配了Heisenberg Double $ \ Mathcal H（G）$的副本。 $γ$的每个顶点（面部）在这些海森伯格双打的产品上定义了$ g $（$ g^*$）的泊松动作。顶点和邻近面的动作形成了双泊松lie组$ d（g）$的泊松动作。我们定义了顶点和面对操作员的泊松对应物，并通过泊松支架将其与vector字段联系起来，从而生成$ d（g）$的动作。 我们在此Poisson-Geometical Kitaev模型与Fock和Rosly的Poisson poisson结构之间构建了Poisson $ d（g）$的同构，用于图$γ$和Poisson-lie组$ d（g）$。这将后者解开，并将其表示为海森堡的产物。它还将Poisson-Geometical Kitaev模型与平面$ d（g）$ - 在定向表面上的模量空间上的象征结构相关联，其边界是从$γ$构建的。
We define Poisson-geometric analogues of Kitaev's lattice models. They are obtained from a Kitaev model on an embedded graph $Γ$ by replacing its Hopf algebraic data with Poisson data for a Poisson-Lie group G. Each edge is assigned a copy of the Heisenberg double $\mathcal H(G)$. Each vertex (face) of $Γ$ defines a Poisson action of $G$ (of $G^*$) on the product of these Heisenberg doubles. The actions for a vertex and adjacent face form a Poisson action of the double Poisson-Lie group $D(G)$. We define Poisson counterparts of vertex and face operators and relate them via the Poisson bracket to the vector fields generating the actions of $D(G)$. We construct an isomorphism of Poisson $D(G)$-spaces between this Poisson-geometrical Kitaev model and Fock and Rosly's Poisson structure for the graph $Γ$ and the Poisson-Lie group $D(G)$. This decouples the latter and represents it as a product of Heisenberg doubles. It also relates the Poisson-geometrical Kitaev model to the symplectic structure on the moduli space of flat $D(G)$-bundles on an oriented surface with boundary constructed from $Γ$.
