Dynamical r matrices and chiral WZNW phase space

A dynamical generalization of the classical Yang-Baxter equation that governs the possible Poisson structures on the space of chiral WZNW fields with a generic monodromy is reviewed. It is explained that, for particular choices of chiral WZNW Poisson brackets, this equation reduces to the CDYB equation recently studied by Etingof and Varchenko and by others. Interesting dynamical r matrices are obtained for a generic monodromy, as well as by imposing Dirac constraints on the monodromy.     

Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems

A new n-dimensional family of Poisson structures is globally characterized and analyzed, including the construction of its main features: the symplectic structure and the reduction to the Darboux canonical form. Examples are given that include the generalization of previously known solution families such as the separable Poisson structures.     

On Quantizable Odd Lie Bialgebras

Motivated by the obstruction to the deformation quantization of Poisson structures in infinite dimensions, we introduce the notion of a quantizable odd Lie bialgebra. The main result of the paper is a construction of the highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so-called quantizable Poisson structures.     

Semiclassical Limits of Ore Extensions and a Poisson Generalized Weyl Algebra

We observe [Launois and Lecoutre, Trans. Am. Math. Soc. 368:755–785, 2016, Proposition 4.1] that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra A₁, considered as a Poisson version of the quantum generalized Weyl algebra, is constructed and its Poisson structures are studied. In particular, a necessary and sufficient condition is obtained, such that A₁ is Poisson simple and established that the Poisson endomorphisms of A₁ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.     

Nambu–Dirac Structures for Lie Algebroids

The theory of Nambu–Poisson structures on manifolds is extended to the context of Lie algebroids in a natural way based on the derived bracket associated with the Lie algebroid differential. A new way of combining Nambu–Poisson structures and triangular Lie bialgebroids is described in this work. Also, we introduce the concept of a higher order Dirac structure on a Lie algebroid. This allows to describe both Nambu–Poisson structures and Dirac structures on manifolds in the same setting.     

On quadratic Poisson structures

We provide a general study on quadratic Poisson structures on a vector space. In particular, we obtain a decomposition for any quadratic Poisson structures. As an application, we classify all the three-dimensional quadratic Poisson structures up to a Poisson diffeomorphism.Research partially supported by NSF Grant DMS 90-01956 and Research Foundation of the University of Pennsylvania.     

Photoluminescence of HgCdTe heterostructures with multiple quantum wells

An analysis of experimental data obtained by various authors via recording infrared (IR) photoluminescence (PL) in heteroepitaxial structures based on solid Cd _x Hg_1?x Te (CHT) solutions with multiple quantum wells is presented. A theoretical analysis of optical transition energies is conducted based on a self-consistent solution to the Poisson and Schröbinger equations for a quantum well. Experimental data obtained at different temperatures, pump powers, and excitation wavelengths are analyzed.     

Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM ⁿ. The central objects in this theory are supplementary invariant Poisson structuresP _c which are incompatable with the original Poisson structureP ₁ for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP ₁ andP ₂ in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP ₁ nor with respect toP ₂. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.Supported by NSERC grant OGPIN 337.     

Gerstenhaber Algebras and BV-Algebras in Poisson Geometry

The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algebras. Some applications are discussed. In particular, we find an explicit connection between the Koszul-Brylinski operator and the modular class of a Poisson manifold. As a consequence, we prove that Poisson homology is isomorphic to Poisson cohomology for unimodular Poisson structures.     

Symmetry Reduction in Symplectic and Poisson Geometry

We present a quick review of several reduction techniques for symplectic and Poisson manifolds using local and global symmetries compatible with these structures. Reduction based on the standard momentum map (symplectic or Marsden–Weinstein reduction) and on generalized distributions (the optimal momentum map and optimal reduction) is emphasized. Reduction of Poisson brackets is also discussed and it is shown how it defines induced Poisson structures on cosymplectic and coisotropic submanifolds.     

Generalized Drinfel'd-Sokolov hierarchies

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct theW _n ^(l) algebras, first discussed for the casen=3 andl=2 by Polyakov and Bershadsky.     

Geometry of Maurer-Cartan Elements on Complex Manifolds

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie algebroid theory. In particular, we extend Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology to the realm of extended Poisson manifolds; we establish a sufficient criterion for these to be finite dimensional; we describe how homology and cohomology are related through the Evens-Lu-Weinstein duality module; and we describe a duality on Koszul-Brylinski homology, which generalizes the Serre duality of Dolbeault cohomology.     

Accurate modelling of InGaN quantum wells

The internal field, the band structure and the oscillator strengths of the optical transitions of wurtzite strained InGaN quantum wells are accurately computed by a self-consistent solution of the Poisson equation and an eight-band k · p Schrödinger equation taking into account charges due to polarisation fields, doping and free carriers. The results are used to compare luminescence and gain spectra for single and triple quantum well structures and to elucidate the effect of the polarisation fields.     

Classical Dynamical r-Matrices¶and Homogeneous Poisson Structures on G/H and K/T

Let G be a finite dimensional simple complex group equipped with the standard Poisson Lie group structure. We show that all G-homogeneous (holomorphic) Poisson structures on G/H, where H⊂G is a Cartan subgroup, come from solutions to the Classical Dynamical Yang–Baxter equations which are classified by Etingof and Varchenko. A similar result holds for a maximal compact subgroup K, and we get a family of K-homogeneous Poisson structures on K/T, where T=K∩H is a maximal torus of K. This family exhausts all K-homogeneous Poisson structures on K/T up to isomorphisms. We study some Poisson geometrical properties of members of this family such as their symplectic leaves, their modular classes, and the moment maps for the T-action. Received: 4 September 1999 / Accepted: 25 January 2000     

Homological properties of certain Generalized Jacobian Poisson Structures in dimension 3

The unimodularity condition for a Poisson structure (i.e., a Poisson structure with a trivial modular class) induces a Poincaré duality between its Poisson homology and its Poisson cohomology. Therefore, information about the Poisson homology of this kind Poisson structures induces by duality information about its Poisson cohomology and vise versa. However, it is not longer true in the case of a non-trivial modular class, which is the case of Generalized Jacobian Poisson Structures (GJPS). In this paper, certain GJPS are considered in dimension 3 and the properties of their Poisson homological groups and their Poisson cohomological groups are obtained. More precisely, under some assumptions, the Poincaré series of these Poisson homological groups are obtained, and these Poisson cohomological groups are computed explicitly, except the second group, which seems to be more complicated to obtain.     

Instantons, Poisson Structures and Generalized Kähler Geometry

Using the idea of a generalized Kähler structure, we construct bihermitian metrics on CP² and CP¹×CP¹, and show that any such structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual connections on a fixed principal bundle over M. We highlight the role of holomorphic Poisson structures in all these constructions.