Poisson-geometric Analogues of Kitaev Models

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 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 .