The covariant Poisson equation for Lie algebra-valued mappings defined in 3-dimensional Euclidean space is studied using functional analytic methods. Weighted covariant Sobolev spaces are defined and used to derive sufficient conditions for the existence and smoothness of solutions to the covariant Poisson equation. These conditions require, apart from suitable continuity, appropriate local integrability of the gauge potentials and global weighted integrability of the curvature form and the source. The possibility of nontrivial asymptotic behaviour of a solution is also considered. As a by-product, weighted covariant generalisations of Sobolev embeddings are established.
A Gauge-Covariant Semiclassical Limit from the magnetic Hartree--Fock equation to the magnetic Vlasov--Poisson equation
Department of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院，教育部数学与复杂系统重点实验室）
The covariant canonical formalism is a covariant extension of the traditional canonical formalism of fields. In contrast to the traditional canonical theory, it has a remarkable feature that canonical equations of gauge theories or gravity are not only manifestly Lorentz covariant but also gauge covariant or diffeomorphism covariant. A mathematical peculiarity of the covariant canonical formalism is that its canonical coordinates are differential forms on a manifold. In the present paper, we find a natural Poisson bracket of this new canonical theory, and study symplectic structure behind it. …
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…
Noticing that the space of the solutions of a first order Hamiltonian field theory has a pre-symplectic structure, we describe a class of conserved charges on it associated to the momentum map determined by any symmetry group of transformations. Gauge theories are dealt with by using a symplectic regularization based on an application of Gotay's coisotropic embedding theorem. The analysis of Electrodynamics and of the Klein-Gordon theory illustrates the main results of the theory as well as the emergence of the energy-momentum tensor algebra of conserved currents.
The formulation of covariant brackets on the space of solutions to a variational problem is analyzed in the framework of contact geometry. It is argued that the Poisson algebra on the space of functionals on fields should be read as a Poisson subalgebra within an algebra of functions equipped with a Jacobi bracket on a suitable contact manifold.
From 2-covariant tensor fields to Poisson and Jacobi structures
Instituto de Ciencias Matemáticas (ICMAT–CSIC)（数学科学研究所）；Real Academia de Ciencias Exactas, Físicas y Naturales de España（西班牙皇家精确、物理和自然科学院）；Great Bay University（大湾区大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；Universitat Rovira i Virgili（罗维拉-伊维尔基里大学）
Many physically important mechanical systems may be described with a Lie group $G$ as configuration space. According to the well-known Noether's theorem, underlying symmetries of the Lie group may be used to considerably reduce the complexity of the problems. However, these reduction techniques, used without care for general problems (waves, field theory), may lead to uncomfortable infinite dimensional spaces. As an alternative, the \emph{covariant} formulation allows to consider a finite dimensional configuration space by increasing the number of independent variables. But the geometric eleme…
The Poisson gauge algebra is a semi-classical limit of complete non-commutative gauge algebra. In the present work we formulate the Poisson gauge theory which is a dynamical field theoretical model having the Poisson gauge algebra as a corresponding algebra of gauge symmetries. The proposed model is designed to investigate the semi-classical features of the full non-commutative gauge theory with coordinate dependent non-commutativity $Θ^{ab}(x)$, especially whose with a non-constant rank. We derive the expression for the covariant derivative of matter field. The commutator relation for the cov…
We analyze higher gauge theories in various dimensions using a supergeometric method based on a differential graded symplectic manifold, called a QP-manifold, which is closely related to the BRST-BV formalism in gauge theories. Extensions of the Lie 2-algebra gauge structure are formulated within the Lie n-algebra induced by the QP-structure. We find that in 5 and 6 dimensions there are special extensions of the gauge algebra. In these cases, a restriction of the gauge symmetry by imposing constraints on the auxiliary gauge fields leads to a covariantized theory. As an example we show that we …
The first part of this paper is devoted to the theory of Poisson-Lie groups in the Banach setting. Our starting point is the straightforward adaptation of the notion of Manin triples to the Banach context. The difference with the finite-dimensional case lies in the fact that a duality pairing between two non-reflexive Banach spaces is necessary weak (as opposed to a strong pairing where one Banach space can be identified with the dual space of the other). The notion of generalized Banach Poisson manifolds introduced in this paper is compatible with weak duality pairings between the tangent spa…
We consider the integrability of a two-parameter deformation of the Wess-Zumino-Witten model, previously introduced in relation with Poisson-Lie T-duality. The resulting family of Poisson-Lie dual models is shown to be integrable by using the Maillet r/s formalism.
We briefly review our results on the Lie theory underlying vector bundles over Lie groupoids and Lie algebroids, pointing out the role of Poisson geometry in extending these results to double Lie algebroids and LA-groupoids.
In this paper, we introduce right-invariant (similarly, left-invariant) Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called $(Λ, \mathbf{n})-$structures. We present a mutual correspondence between $(Λ,\mathbf{n})-$structures on Lie algebroids with Poisson-Nijenhuis structures $(Π, \mathbf{N})$ on their Lie groupoids under some conditions. Also, we will construct Poisson-Nijenhuis pair groupoids as an example from the invariance's point of view.
New York University Abu Dhabi（阿布扎比纽约大学）；NYUAD Research Institute（NYUAD研究院）；Center for Quantum and Topological Systems, NYUAD Research Institute（NYUAD研究院量子与拓扑系统中心）
We introduce a dual notion of the Poisson algebra by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. We show that the transposed Poisson algebra thus defined not only shares common properties of the Poisson algebra, including the closure under taking tensor products and the Koszul self-duality as an operad, but also admits a rich class of identities. More significantly, a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. Consequently, the classic constructio…
Considering a Poisson algebra as a non associative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this non associative algebra. This gives a natural interpretation of deformations which preserves the underlying associative structure and we study deformations which preserve the underlying Lie algebra.
We investigate some infinite dimensional Lie algebras and their associated Poisson structures which arise from a Lie group action on a manifold. If $G$ is a Lie group, $\g$ its Lie algebra and $M$ is a manifold on which $G$ acts, then the set of smooth maps from $M$ to $\g$ has at least two Lie algebra structures, both satisfying the required property to be a Lie algebroid. We may then apply a {construction} by Marle to obtain a Poisson bracket on the set of smooth real or complex valued functions on $M\times \g^*$. In this paper, we investigate these Poisson brackets. We show that the set of …
In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson-Lie structure preserve a multiple of any left-invariant volume on the group. Conversely, we also prove that if there exists a Hamiltonian function such that the identity element of the Lie group is a nondegenerate singularity and the associated Hamiltonian vector field preserves a volume form, then the Po…
In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a quadratic Lie algebra. We prove a uniqueness statement about these Poisson brackets with a given moment map. As an application we get formulae for the linearization of the quasi-Poisson structure of the moduli space of flat connections on a punctured sphere, and thereby identify their symplectic leaves with the reduction of coadjoint orbits. Equivalently, we …
For a particular class of backgrounds, equations of motion for string sigma models targeted in mutually dual Poisson-Lie groups are equivalent. This phenomenon is called the Poisson-Lie T-duality. On the level of the corresponding string effective actions, the situation becomes more complicated due to the presence of the dilaton field. A novel approach to this problem using Levi-Civita connections on Courant algebroids is presented. After the introduction of necessary geometrical tools, formulas for the Poisson-Lie T-dual dilaton fields are derived. This provides a version of Poisson-Lie T-dua…
We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.
We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished nilpotent elements of semisimple type.
