Alberto S. Cattaneo, Ping Xu
Poisson manifolds may be regarded as the infinitesimal form of symplectic groupoids. Twisted Poisson manifolds considered by Severa and Weinstein [math.SG/0107133] are a natural generalization of the former which also arises in string theory. In this note it is proved that twisted Poisson manifolds are in bijection with a (possibly singular) twisted version of symplectic groupoids.
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We develop a theory of quasi-Lie bialgebroids using a homological approach. This notion is a generalization of quasi-Lie bialgebras, as well as twisted Poisson structures with a 3-form background which have recently appeared in the context of string theory, and were studied by Ševera and Weinstein using a different method.
We study the geometric and algebraic properties of the twisted Poisson structures on Lie algebroids, leading to a definition of their modular class and to an explicit determination of a representative of the modular class, in particular in the case of a twisted Poisson manifold.
We derive a formula for the the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.
Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in \cite{balseiro_garcia_naranjo}, this paper investigates the global theory of integrable Hamiltonian systems on almost symplectic manifolds as an initial step to understand Hamiltonian integrability on twisted Poisson (and Dirac) manifolds. Non-commutative integrable Hamiltonian systems on almost symplectic manifolds were first defined in \cite{fasso_sansonetto}, which proved existence of local generalised action-angle coordinates in the spirit of the Liouville-Arnol'd theorem. In an…
For a particle in the magnetic field of a cloud of monopoles, the naturally associated 2-form on phase space is not closed, and so the corresponding bracket operation on functions does not satisfy the Jacobi identity. Thus, it is not a Poisson bracket; however, it is twisted Poisson in the sense that the Jacobiator comes from a closed 3-form. The space $\mathcal D$ of densities on phase space is the state space of a plasma. The twisted Poisson bracket on phase-space functions gives rise to a bracket on functions on $\mathcal D$. In the absence of monopoles, this is again a Poisson bracket. It …
We describe a deformation quantization of a modification of Poisson geometry by a closed 3-form. Under suitable conditions it gives rise to a stack of algebras. The basic object used for this aim is a kind of families of Poisson structures given by a Maurer-Cartan equation; they are easily quantized using Kontsevich's Formality theorem.
This paper is devoted to the study of Morita equivalence for twisted Poisson manifolds. We review some Morita invariants and prove that integrable twisted Poisson manifolds which are gauge equivalent are Morita equivalent. Moreover, we introduce the notion of weak Morita equivalence and show that if two twisted Poisson manifolds are weak Morita equivalent, there exists a one-to-one correspondence between their twisted symplectic leaves.
We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a "generalized" Rota-Baxter identity of weight zero. Such operators are called generalized Rota-Baxter operators. We will show that generalized Rota-Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator "twisted Rota-Baxter operators" which is a natural generalization of generalized Rota-Baxter operators. It is known that classical Rota-Baxter operators ar…
We define algebras of admissible functions associated to twisted Dirac structures, and we show that they are Poisson algebras. We study the standard cases associated to Dirac structures defined by graphs of non-degenerate 2-forms.
Quadratic Poisson tensors of the Dufour-Haraki classification read as a sum of an $r$-matrix induced structure twisted by a (small) compatible exact quadratic tensor. An appropriate bigrading of the space of formal Poisson cochains then leads to a vertically positive double complex. The associated spectral sequence allows to compute the Poisson-Lichnerowicz cohomology of the considered tensors. We depict this modus operandi, apply our technique to concrete examples of twisted Poisson structures, and obtain a complete description of their cohomology. As richness of Poisson cohomology entails co…
We construct an action of the braid group B_N on the twisted quantized enveloping algebra U'_q(o_N) where the elements of B_N act as automorphisms. In the classical limit q -> 1 we recover the action of B_N on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and re-discovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twiste…
In this paper we study the relationship between the extended symmetries of exact Courant algebroids over a manifold $M$, defined by Bursztyn, Cavalcanti and Gualtieri, and the Poisson algebras of admissible functions associated to twisted Dirac structures when acted by Lie groups. We show that the usual homomorphisms of Lie algebras between the algebras of infinitesimal symmetries of the action, vector fields on the manifold and the Poisson algebra of observables, appearing in symplectic geometry, generalize to natural maps of Leibniz algebras induced both by the extended action and compatible…
Motivated by questions from quantum group and field theories, we review structures on manifolds that are weaker versions of Poisson structures, and variants of the notion of Lie algebroid. We give a simple definition of the Courant algebroids and introduce the notion of a deriving operator for the Courant bracket of the double of a proto-bialgebroid. We then describe and relate the various quasi-Poisson structures, which have appeared in the literature since 1991, and the twisted Poisson structures studied by Severa and Weinstein.
After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.
We study a modification of Poisson geometry by a closed 3-form. Just as for ordinary Poisson structures, these "twisted" Poisson structures are conveniently described as Dirac structures in suitable Courant algebroids. The additive group of 2-forms acts on twisted Poisson structures and permits them to be seen as glued from ordinary Poisson structures defined on local patches. We conclude with remarks on deformation quantization and twisted symplectic groupoids.
We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic groupoids with twisted contact ones. Some examples for each structure are presented.
Let $G$ be a connected complex semisimple Lie group with a fixed maximal torus $T$ and a Borel subgroup $B \supset T$. For an arbitrary automorphism $θ$ of $G$, we introduce a holomorphic Poisson structure $π_θ$ on $G$ which is invariant under the $θ$-twisted conjugation by $T$ and has the property that every $θ$-twisted conjugacy class of $G$ is a Poisson subvariety with respect to $π_θ$. We describe the $T$-orbits of symplectic leaves, called $T$-leaves, of $π_θ$ and compute the dimensions of the symplectic leaves (i.e, the ranks) of $π_θ$. We give the lowest rank of $π_θ$ in any given $θ$-t…
In recent years methods for the integration of Poisson manifolds and of Lie algebroids have been proposed, the latter being usually presented as a generalization of the former. In this note it is shown that the latter method is actually related to (and may be derived from) a particular case of the former if one regards dual of Lie algebroids as special Poisson manifolds. The core of the proof is the fact, discussed in the second part of this note, that coisotropic submanifolds of a (twisted) Poisson manifold are in one-to-one correspondence with possibly singular Lagrangian subgroupoids of sou…
Let $B$ be a twisted Poisson manifold with a fixed tropical affine structure given by a period bundle $P$. In this paper, we study the classification of almost symplectically complete isotropic realizations (ASCIRs) over $B$ in the spirit of \cite{DD}. We construct a product among ASCIRs in analogy with tensor product of line bundles, thereby introducing the notion of the Picard group of $B$. We give descriptions of the Picard group in terms of exact sequences involving certain sheaf cohomology groups, and find that the `Néron-Severi group' is isomorphic to $H^2(B, \underline{P})$. An example …
The correspondence between Poisson structures and symplectic groupoids, analogous to the one of Lie algebras and Lie groups, plays an important role in Poisson geometry; it offers, in particular, a unifying framework for the study of hamiltonian and Poisson actions. In this paper, we extend this correspondence to the context of Dirac structures twisted by a closed 3-form. More generally, given a Lie groupoid $G$ over a manifold $M$, we show that multiplicative 2-forms on $G$ relatively closed with respect to a closed 3-form $ϕ$ on $M$ correspond to maps from the Lie algebroid of $G$ into the c…
A nonzero 2-cocycle $Γ\in Z^2(\g,\R)$ on the Lie algebra $\g$ of a compact Lie group $G$ defines a twisted version of the Lie-Poisson structure on the dual Lie algebra $\g^*$, leading to a Poisson algebra $C^{\infty}(\g_{(Γ)}^*)$. Similarly, a multiplier $c\in Z^2(G,U(1))$ on $G$ which is smooth near the identity defines a twist in the convolution product on $G$, encoded by the twisted group $C^*$-algebra C^*(G,c)$. Further to some superficial yet enlightening analogies between $C^{\infty}(\g^*_{(Γ)})$ and $C^*(G,c)$, it is shown that the latter is a strict quantization of the former, where Pl…
In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on $C^{N}$ with the property that for any $n,m\in N$ such that $n m =N$, the restriction of the Poisson algebra to the space of bilinear forms with block-upper-triangular matrix composed from blocks of size $m\times m$ is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-upper-triangular bilinear forms. The groupoid elements automatically preser…
Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds
本文在（预）多重辛流形上的李代数胚中引入相容 E-n-形式，统一刻画扭曲泊松结构、动量映射及其同伦推广等结构。
In this paper we study quotients of Lie algebroids and groupoids endowed with compatible differential forms. We identify Lie theoretic conditions under which such forms become basic and characterize the induced forms on the quotients. We apply these results to describe generalized quotient and reduction processes for (twisted) Poisson and Dirac structures, as well as to their integration by (twisted, pre-)symplectic groupoids. In particular, we recover and generalize several known results concerning Poisson reduction.
Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal bundle $M\to M/H$, integrations of a Dirac structure on $M/H$ to $H$-admissible integrations of its pullback Dirac structure on $M$ by pre-symplectic groupoids. Our construction gives a distinguished class of explicit real or holomorphic pre-symplectic and symplectic groupoids over semi-simple Lie groups and some of their homogeneou…
We extend the correspondence between Poisson maps and actions of symplectic groupoids, which generalizes the one between momentum maps and hamiltonian actions, to the realm of Dirac geometry. As an example, we show how hamiltonian quasi-Poisson manifolds fit into this framework by constructing an ``inversion'' procedure relating quasi-Poisson bivectors to twisted Dirac structures.
The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons (a twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation). We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynami…
Integration of the elliptic tangent bundle and elliptic Poisson structures
本文显式构造了整合椭圆切丛的Lie群胚，给出了Hausdorff整合的充要拓扑条件，并为椭圆Poisson结构的辛整合提供了显式局部模型。
We study a reduction procedure for describing the symplectic groupoid of a Poisson homogeneous space obtained by quotient of a coisotropic subgroup. We perform it as a reduction of the Lu-Weinstein symplectic groupoid integrating Poisson Lie groups, that is suitable even for the non complete case.
We introduce the notion of twisted generalized complex submanifolds and describe an equivalent characterization in terms of Poisson-Dirac submanifolds. Our characterization recovers a result of Vaisman. An equivalent characterization is also given in terms of spinors. As a consequence, we show that the fixed locus of an involution preserving a twisted generalized complex structure is a twisted generalized complex submanifold. We also prove that a twisted generalized complex manifold has a natural Poisson structure. We also discuss generalized Kaehler submanifolds.
We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on a special class called $b$-Poisson/$b$-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle coordinate theorems in these settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds [LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a $b$-Poisson manifold…
We consider the problem of deforming simultaneously a pair of given structures. We show that such deformations are governed by an L-infinity algebra, which we construct explicitly. Our machinery is based on Th. Voronov's derived bracket construction. In this paper we consider only geometric applications, including deformations of coisotropic submanifolds in Poisson manifolds, of twisted Poisson structures, and of complex structures within generalized complex geometry. These applications can not be, to our knowledge, obtained by other methods such as operad theory.
We describe three perspectives on higher quantization, using the example of magnetic Poisson structures which embody recent discussions of nonassociativity in quantum mechanics with magnetic monopoles and string theory with non-geometric fluxes. We survey approaches based on deformation quantization of twisted Poisson structures, symplectic realization of almost symplectic structures, and geometric quantization using 2-Hilbert spaces of sections of suitable bundle gerbes. We compare and contrast these perspectives, describing their advantages and shortcomings in each case, and mention many ope…
We provide an explicit description of symplectic leaves of a simply connected connected semisimple complex Lie group equipped with the standard Poisson-Lie structure. This sharpens previously known descriptions of the symplectic leaves as connected components of certain varieties. Our main tool is the machinery of twisted generalized minors. They also allow us to present several quasi-commuting coordinate systems on every symplectic leaf. As a consequence, we construct new completely integrable systems on some special symplectic leaves.
Symplectic and Poisson structures of certain moduli spaces/Huebschmann,J./ Abstract: Let $π$ be the fundamental group of a closed surface and $G$ a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a certain construction due to A. Weinstein relying on techniques from equivariant cohomology may be refined so as to yield (i) a symplectic structure on a certain smooth manifold $\Cal M(\Cal P,G)$ containing the space $\roman{Hom}(π,G)$ of homomorphisms and, furthermore, (ii) a hamiltonian $G$-action on $\Cal M(\Cal P,G)$ preserving the symplectic structure, w…
Integration and quantization of 1-shifted coisotropics
本文将 Cattaneo–Felder 的余等值约化积分与量子化框架推广到 1-平移辛可微层中的 1-平移余等值面，建立积分定理并构造平坦 P_∞ 结构与弯曲 A_∞ 量子化，恢复约化泊松括号。
These notes discuss various aspect of the ``representation theory'' of Poisson manifolds, with focus on Morita equivalence and Picard groups. We give a brief introduction to Poisson geometry (including Dirac and twisted Poisson structures) and algebraic Morita theory before presenting the geometric Morita theory of Poisson manifolds. We also point out the connections with the theory of symplectic groupoids and hamiltonian actions.
Motivated by a search for Lie group structures on groups of Poisson diffeomorphisms [24], we investigate linearizability of Poisson structures of Poisson groupoids around the unit section. After extending the Lagrangian neighbourhood theorem to the setting of cosymplectic Lie algebroids, we establish that dual integrations of triangular bialgebroids are always linearizable. Additionally, we show that the (non-dual) integration of a triangular Lie bialgebroid is linearizable whenever the $r$-matrix is of so-called cosymplectic type. The proof relies on the integration of a triangular Lie bialge…
We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express the obstructions in terms of variations of symplectic areas. As an application of these results, we show that a Poisson manifold admits a complete symplectic realization if, and only if, it is integrable. We discuss also the integration of submanifolds and Morita equivalence of Poi…
In this note a functorial approach to the integration problem of an LA-groupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibred products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns out, that the fibred product of Lie algebroids along integrable morphisms is always integrable by a fibred product of Lie groupoids. We show that to every LA-groupoid with integrable top structure one can associate a differentiable graph in the category of Lie groupoids, which is…
We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian action on the symplectic groupoid $Σ(M)$. We give applications of these results to the integration of Poisson quotients $M/G$, Lu-Weinstein quotients $μ^{-1}(e)/G$ and Poisson homogeneous spaces $G/H$.
In this paper, we consider the hamiltonian formulation of nonholonomic systems with symmetries and study several aspects of the geometry of their reduced almost Poisson brackets, including the integrability of their characteristic distributions. Our starting point is establishing global formulas for the nonholonomic Jacobiators, before and after reduction, which are used to clarify the relationship between reduced nonholonomic brackets and twisted Poisson structures. For certain types of symmetries (generalizing the Chaplygin case), we obtain genuine Poisson structures on the reduced spaces an…
We evaluate the path integral of the Poisson sigma model on sphere and study the correlators of quantum observables. We argue that for the path integral to be well-defined the corresponding Poisson structure should be unimodular. The construction of the finite dimensional BV theory is presented and we argue that it is responsible for the leading semiclassical contribution. For a (twisted) generalized Kahler manifold we discuss the gauge fixed action for the Poisson sigma model. Using the localization we prove that for the holomorphic Poisson structure the semiclassical result for the correlato…
Quasi-Poisson varieties from double quasi-Poisson algebras in types $B,C,D$
本文通过引入对合反自同构，将双拟泊松括号推广到任意半单基上的扭表示空间，并应用于箭图以理解扭（局部乘性）箭图簇上的泊松结构，同时恢复了正交/辛群特征簇的泊松结构并引入修正的Kontsevich系统。
Coupling Dirac structures are Dirac structures defined on the total space of a fibration, generalizing hamiltonian fibrations from symplectic geometry, where one replaces the symplectic structure on the fibers by a Poisson structure. We study the associated Poisson gauge theory, in order to describe the presymplectic groupoid integrating coupling Dirac structures. We find the obstructions to integrability and we give explicit geometric descriptions of the integration.
We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the category of integrable Lie algebroids and complete Lie algebroid comorphisms to the category of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of comorphisms and Poisson maps may not hold in the absence of a completeness assumption.
Shifted lagrangian structures in Poisson geometry
本文通过建立移位辛几何与经典泊松几何之间的新联系，利用2-移位辛群中的拉格朗日态射，发展了狄拉克结构与拟泊松流形积分的理论。
We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these objects are related by differentiation and integration. We use these results to revisit Lie 2-bialgebras by means of Poisson double structures.
We use the techniques of integration of Poisson manifolds into symplectic Lie groupoids to build symplectic resolutions (= desingularizations) of the closure of a symplectic leaf. More generally, we show how Lie groupoids can be used to lift singularities, in particular when one imposes a compatibility condition with an additional structure given by a multi-vector field.
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