Poisson-geometric analogues of Kitaev models. (English) Zbl 1464.81050
The author constructs a Poisson analogue of the Kitaev model and relates it to the Poisson structure introduced by Fock and Rosly and to the moduli spaces of flat \(G\)-bundles [V. V. Fock and A. A. Rosly, Transl., Ser. 2, Am. Math. Soc. 191, 67–86 (1999; Zbl 0945.53050)]. The author shows that the moduli space for a surface with boundary can be viewed as a Poisson counterpart to a Kitaev model with excitations, or quasi-particles, located at the boundary components. These results also allow for a decoupling of the symplectic structure on the moduli space by describing it in terms of a product Poisson manifold associated to a Poisson-geometrical Kitaev model.
As described in the introduction of the paper under review, the Poisson-Lie groups can be regarded as Poisson analogues of Hopf algebras and many structures and constructions for Hopf algebras have Poisson-Lie group counterparts. Like Hopf algebras, each Poisson-Lie group \(G\) possesses a dual Poisson-Lie group \(G^*\), and there are Poisson-geometrical notions of Heisenberg and Drinfeld doubles. Additionally, a counterpart to a module algebra over a Hopf algebra is given by the Poisson algebra \(C^\infty(M,\mathbb{R})\) of functions on a Poisson \(G\)-space \(M\), which is a Poisson manifold \(M\) together with a Poisson action of the Poisson-Lie group \(G\).
By using this correspondence the author defines an analogue of Kitaev models by replacing the Hopf-algebraic data of a Kitaev model by its Poisson-Lie counterparts. For this purpose, the author considers an embedded graph \(\Gamma\) with edge set \(E\) and assigns a copy of the Heisenberg double \(\mathcal{H}(G)\) of a Poisson-Lie group \(G\) to every edge to obtain the product Poisson manifold \(\mathcal{H}(G)^{\times E}\). The Poisson algebra of functions \(C^\infty\left(\mathcal{H}(G)^{\times E} ,\mathbb{R}\right)\) is the counterpart to the endomorphism algebra of the extended space \(\mathrm{End}_{\mathbb{C}}(H^{\otimes E})\cong\mathcal{H}(H)^{\otimes E}\), where \(H\) denotes a finite-dimensional semi-simple Hopf \(*\)-algebra over \(\mathbb{C}\), [C. Meusburger, Commun. Math. Phys. 353, No. 1, 413–468 (2017; Zbl 1460.81064)].
As the author explains, this work can be completed by writing a Poisson analogue of the Hamiltonian of a Kitaev model, constructing an analogue of the ground state and a thorough investigation of the Poisson analogues of the ribbon operators.
As described in the introduction of the paper under review, the Poisson-Lie groups can be regarded as Poisson analogues of Hopf algebras and many structures and constructions for Hopf algebras have Poisson-Lie group counterparts. Like Hopf algebras, each Poisson-Lie group \(G\) possesses a dual Poisson-Lie group \(G^*\), and there are Poisson-geometrical notions of Heisenberg and Drinfeld doubles. Additionally, a counterpart to a module algebra over a Hopf algebra is given by the Poisson algebra \(C^\infty(M,\mathbb{R})\) of functions on a Poisson \(G\)-space \(M\), which is a Poisson manifold \(M\) together with a Poisson action of the Poisson-Lie group \(G\).
By using this correspondence the author defines an analogue of Kitaev models by replacing the Hopf-algebraic data of a Kitaev model by its Poisson-Lie counterparts. For this purpose, the author considers an embedded graph \(\Gamma\) with edge set \(E\) and assigns a copy of the Heisenberg double \(\mathcal{H}(G)\) of a Poisson-Lie group \(G\) to every edge to obtain the product Poisson manifold \(\mathcal{H}(G)^{\times E}\). The Poisson algebra of functions \(C^\infty\left(\mathcal{H}(G)^{\times E} ,\mathbb{R}\right)\) is the counterpart to the endomorphism algebra of the extended space \(\mathrm{End}_{\mathbb{C}}(H^{\otimes E})\cong\mathcal{H}(H)^{\otimes E}\), where \(H\) denotes a finite-dimensional semi-simple Hopf \(*\)-algebra over \(\mathbb{C}\), [C. Meusburger, Commun. Math. Phys. 353, No. 1, 413–468 (2017; Zbl 1460.81064)].
As the author explains, this work can be completed by writing a Poisson analogue of the Hamiltonian of a Kitaev model, constructing an analogue of the ground state and a thorough investigation of the Poisson analogues of the ribbon operators.
Reviewer: Farhang Loran (Isfahan)
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 53D50 | Geometric quantization |
| 81S10 | Geometry and quantization, symplectic methods |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 58D27 | Moduli problems for differential geometric structures |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 16T05 | Hopf algebras and their applications |
| 17B63 | Poisson algebras |
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