Poisson Structures and Potentials

Contemporary Mathematics, 1984

A global formula for Poisson brackets on reduced cotangent bundles of principal bundles is derived. The result bears on the basic constructions for interacting systems due to Sternberg and Weinstei n and on Poi,sson brackets i nvol ving semi-direct products for fluid and plasma systems. The formula involves Lie-Poisson structures, canonical brackets, and curvature terms. 1. INTRODUCTION. Let G be a Lie group and tJd its Lie algebra. The right (resp. left) reduction of T*G by G produces the + (resp.-) Lie-Poisson structure on 0(*: This construction is now well-known and has been reviewed in the lectures of Weinstein, Ratiu and Morrison in these proceedings. This paper concerns the Poisson structure on the reduction of T*B, where 1T:B-.. X is a principal bundle. The Poisson structure on the (right) reduced space G\T*B is a mixture of Lie-Poisson and canonical structures and will be computed explicitly. There are several motivations for considering the constructions presented here. First of all, these reduced spaces occur in the construction of phase spaces for interacting systems: see Sternberg [1977J and Weinstein [1978J for a particle in a Yang-Mills field and Marsden and t-Jeinsten [1982J for the Maxsell-Vlasovequation. The link between the approaches of Sternberg and Weinstein and the physicist's equations (Wong's equations) was given in Montgomery [1983J and provides a basis for this paper. The second motivation was to better understand the role of Lie-Poisson structures associated with semi-direct products of groups G ~ H. The way these arise in examples was first systematically explored by Guillemin and

Poisson-Lie groups and complete integrability. I. Drinfeld bigebras, dual extensions and their canonical representations Annales de l'I. H. P., section A, tome 49, n o 4 (1988), p. 433-460 <http://www.numdam.org/item?id=AIHPA_1988__49_4_433_0> © Gauthier-Villars, 1988, tous droits réservés. L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

The Many Facets of Geometry, 2010

We first extend the notion of connection in the context of Courant algebroids to obtain a new characterization of generalized Kähler geometry. We then establish a new notion of isomorphism between holomorphic Poisson manifolds, which is nonholomorphic in nature. Finally we show an equivalence between certain configurations of branes on Poisson varieties and generalized Kähler structures, and use this to construct explicitly new families of generalized Kähler structures on compact holomorphic Poisson manifolds equipped with positive Poisson line bundles (e.g. Fano manifolds). We end with some speculations concerning the connection to noncommutative algebraic geometry.

Journal of Mathematics Research, 2017

We define the notion of logarithmic Poisson structure along a non zero ideal $\cali$ of an associative, commutative algebra $\cal A$ and prove that each logarithmic Poisson structure induce a skew symmetric 2-form and a Lie-Rinehart structure on the $\cal A$-module $\Omega_K(\log \cali)$ of logarithmic K\&quot;{a}hler differential. This Lie-Rinehart structure define a representation of the underline Lie algebra. Applying the machinery of Chevaley-Eilenberg and Palais, we define the notion of logarithmic Poisson cohomology which is a measure obstructions of Linear representation of the underline Lie algebra for which the grown ring act by multiplication.

arXiv (Cornell University), 2019

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 × g *. In this paper, we investigate these Poisson brackets. We show that the set of examples include the standard Darboux symplectic structure and the classical Lie Poisson brackets, but is a strictly larger class of Poisson brackets than these. Our study includes the associated Hamiltonian flows and their invariants, canonical maps induced by the Lie group action, and compatible Poisson structures. Our approach is mainly computational and we detail numerous examples. The Lie brackets from which our results derive, arose from the consideration of connections on bundles with zero curvature and constant torsion. We give an alternate derivation of the Lie bracket which will be suited to applications to Lie group actions for applications not involving a Riemannian metric. We also begin a study of the infinite dimensional Poisson brackets which may be obtained by considering a central extension of the Lie algebras.

Advances in Mathematics, 2017

For a connected abelian Lie group T acting on a Poisson manifold (Y, π) by Poisson isomorphisms, the T-leaves of π in Y are, by definition, the orbits of the symplectic leaves of π under T, and the leaf stabilizer of a T-leaf is the subspace of the Lie algebra of T that is everywhere tangent to all the symplectic leaves in the T-leaf. In this paper, we first develop a general theory on T-leaves and leaf stabilizers for a class of Poisson structures defined by Lie bialgebra actions and quasitriangular r-matrices. We then apply the general theory to four series of holomorphic Poisson structures on products of flag varieties and related spaces of a complex semi-simple Lie group G. We describe their T-leaf decompositions, where T is a maximal torus of G, in terms of (open) extended Richardson varieties and extended double Bruhat cells associated to conjugacy classes of G, and we compute their leaf stabilizers and the dimension of the symplectic leaves in each T-leaf.

arXiv: Differential Geometry, 2016

For a connected complex semi-simple Lie group $G$, we consider the Poisson structure $\pi_n$ on an $n$-dimensional Bott-Samelson variety $Z_{\bf u}$ of $G$ defined by a standard multiplicative Poisson structure $\pi_{\rm st}$ on $G$, where ${\bf u}$ is any sequence of length $n$ of simple reflections in the Weyl group of $G$. We explicitly express $\pi_n$ on each of the $2^n$ affine coordinate charts, one for every subexpression of ${\bf u}$, in terms of the root strings and the structure constants of the Lie algebra of $G$. We show that the restriction of $\pi_n$ to each affine coordinate chart gives rise to a Poisson structure on the polynomial algebra ${\mathbb{C}}[z_1, \ldots, z_n]$ which is a Poisson-Ore extension of $\mathbb{C}$ compatible with a rational action by a maximal torus of $G$. For canonically chosen $\pi_{\rm st}$, we show that the induced Poisson structure on ${\mathbb{C}}[z_1, \ldots, z_n]$ for every affine coordinate chart is in fact defined over $\mathbb{Z}$, t...

Contemporary Mathematics, 2008

We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H, and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra associated to G/H. We also construct a Poisson groupoid over (G/H, π) which is symplectic near the identity section. This note serves as preparation for forthcoming papers, in which we will compute explicitly the Poisson cohomology and study their symplectic groupoids for certain examples of Poisson homogeneous spaces related to semi-simple Lie groups.

2021

Let $G$ be a connected semisimple Lie group. There are two natural duality constructions that assign to it the Langlands dual group $G^\vee$ and the Poisson-Lie dual group $G^*$. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein-Kazhdan potential on the double Bruhat cell $G^{\vee; w_0, e} \subset G^\vee$ is isomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure on the partial tropicalization of $K^* \subset G^*$ (the Poisson-Lie dual of the compact form $K \subset G$). By [5], the first cone parametrizes the canonical bases of irreducible $G$-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of $K^*$ are eq...

Journal of Geometry and Physics, 1998

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin-and Gauß-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail. 1991 Mathematics Subject Classification. 22E30, 58F05, 70H99.