Poisson-geometric Analogues of Kitaev Models

Language
en
Document Type
Article
Issue Date
2023-03-27
First published
2021-04-01
Issue Year
2021
Authors
Spies, Alexander
Editor
Publisher
Springer Berlin Heidelberg
Abstract

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.

Journal Title
Communications in Mathematical Physics
Volume
383
Issue
1
Citation
Communications in Mathematical Physics 383.1 (2021): S. 345-400. <https://link.springer.com/article/10.1007/s00220-021-03992-5>
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