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 $Γ$.
In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ON-structures on bimodules over pre-Lie algebras. We show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. We study solutions of the strong Maurer-Cartan equation on the twilled pre-Lie algebra associated to an O-operator, which gives rise to a pair of ON-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra g are corresponding to ON-structures on the bimodule $(\mathfrak g^*;\mathrm{ad}^*,-R^*)$, and $KVΩ$-structures …
We present generalizations of the well-known trigonometric spin Sutherland models, which were derived by Hamiltonian reduction of `free motion' on cotangent bundles of compact simple Lie groups based on the conjugation action. Our models result by reducing the corresponding Heisenberg doubles with the aid of a Poisson-Lie analogue of the conjugation action. We describe the reduced symplectic structure and show that the `reduced main Hamiltonians' reproduce the spin Sutherland model by keeping only their leading terms. The solutions of the equations of motion emerge from geodesics on the compac…
The Poisson-Hopf analogue of an arbitrary quantum algebra U_z(g) is constructed by introducing a one-parameter family of quantizations U_{z,h}(g) depending explicitly on h and by taking the appropriate h -> 0 limit. The q-Poisson analogues of the su(2) algebra are discussed and the novel su_q^P (3) case is introduced. The q-Serre relations are also extended to the Poisson limit. This approach opens the perspective for possible applications of higher rank q-deformed Hopf algebras in semiclassical contexts.
In this paper we introduce a kind of "noncommutative neighbourhood" of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a "distortion" of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue o…
We consider odd Poisson (odd symplectic) structure on supermanifolds induced by an odd symmetric rank $2$ (non-degenerate) contravariant tensor field. We describe the difference between odd Riemannian and odd symplectic structure in terms of the Cartan prolongation of the corresponding Lie algebras, and formulate an analogue of the Levi-Civita theorem for an odd symplectic supermanifold.
In this paper, we introduce the notion of an omni $n$-Lie algebra and show that they are linearization of higher analogues of standard Courant algebroids. We also introduce the notion of a nonabelian omni $n$-Lie algebra and show that they are linearization of higher analogues of Courant algebroids associated to Nambu-Poisson manifolds.
A systematic computational approach for the explicit construction of any quantum Hopf algebra (U_z(g),Δ_z) starting from the Lie bialgebra (g,δ) that gives the first-order deformation of the coproduct map Δ_z is presented. The procedure is based on the fact that any quantum algebra can be viewed as the quantization of the unique Poisson-Lie structure (G^\ast,Λ_g) on the dual group G^\ast, which is obtained by exponentiating the Lie algebra g^\ast defined by the dual map δ^\ast. From this perspective, the coproduct for U_z(g) is just the pullback of the group law for G^\ast, and the Poisson ana…
In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle $TM\oplus\wedge^nT^*M$ for an $m$-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an $(n+1)$-vector field $π$ is closed under the higher-order Dorfman bracket iff $π$ is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on $\wedge^nT^*M$. The graph of an $(n+1)$-form $ω$ is closed under the higher-order Dorfman bracket iff $ω$ is a premultis…
We construct an analogue of Dirac's reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac's reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld-Sokolov hierarchy.
We introduce Kähler-Poisson algebras as analogues of algebras of smooth functions on Kähler manifolds, and prove that they share several properties with their classical counterparts on an algebraic level. For instance, the module of inner derivations of a Kähler-Poisson algebra is a finitely generated projective module, and allows for a unique metric and torsion-free connection whose curvature enjoys all the classical symmetries. Moreover, starting from a large class of Poisson algebras, we show that every algebra has an associated Kähler-Poisson algebra constructed as a localization. At the e…
We introduce the notion of Poisson superbialgebra as an analogue of Drinfeld's Lie superbialgebras. We extend various known constructions dealing with representations on Lie superbialgebras to Poisson superbialgebras. We introduce the notions of Manin triple of Poisson superalgebras and Poisson superbialgebras and show the equivalence between them in terms of matched pairs of Poisson superalgebras. A combination of the classical Yang-Baxter equation and the associative Yang-Baxter equation is discussed in this framework. Moreover, we introduce notions of $\mathcal{O}$-operator of weight $λ\in\…
Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie-Hamilton systems, to devise a novel formalism: the Poisson-Hopf algebra deformations of Lie-Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie-Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie-Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie-Hamilton system associated with a Poisson-Hopf algebra of functions that allows for the explicit description of its $t$-independent constant…
In this note we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan-Schwinger (J.-S.) map which has been known and used for a long time by physicists. The difference, comparing to J.-S. map, is that we use generators of Cuntz algebra $\mathcal{O}_{\infty}$ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a "building blocks" instead of creation-annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization i.e. exact conservation of Lie brac…
An analogue of geometric quantization of Poisson algebras obtained by algebraic reduction of symmetries is developed. Interpretation of the obtained results and their application to the problem of commutativity of quantization and reduction are given
We derive the $2$d Zakharov-Mikhailov action from $4$d Chern-Simons theory. This $2$d action is known to produce as equations of motion the flatness condition of a large class of Lax connections of Zakharov-Shabat type, which includes an ultralocal variant of the principal chiral model as a special case. At the $2$d level, we determine for the first time the covariant Poisson bracket $r$-matrix structure of the Zakharov-Shabat Lax connection, which is of rational type. The flatness condition is then derived as a covariant Hamilton equation. We obtain a remarkable formula for the covariant Hami…
The quantum duality principle is used to obtain explicitly the Poisson analogue of the kappa-(A)dS quantum algebra in (3+1) dimensions as the corresponding Poisson-Lie structure on the dual solvable Lie group. The construction is fully performed in a kinematical basis and deformed Casimir functions are also explicitly obtained. The cosmological constant $Λ$ is included as a Poisson-Lie group contraction parameter, and the limit $Λ\to 0$ leads to the well-known kappa-Poincaré algebra in the bicrossproduct basis. A twisted version with Drinfel'd double structure of this kappa-(A)dS deformation i…
A deformation of the classical trigonometric BC(n) Sutherland system is derived via Hamiltonian reduction of the Heisenberg double of SU(2n). We apply a natural Poisson-Lie analogue of the Kazhdan-Kostant-Sternberg type reduction of the free particle on SU(2n) that leads to the BC(n) Sutherland system. We prove that this yields a Liouville integrable Hamiltonian system and construct a globally valid model of the smooth reduced phase space wherein the commuting flows are complete. We point out that the reduced system, which contains 3 independent coupling constants besides the deformation param…
