Poisson Groupoid Research Articles

Configuration Poisson Groupoids of Flags

Abstract Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in ${\mathcal {B}}^n$. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids

We show that the 2d Poisson Sigma Model on a Poisson groupoid arises as an effective theory of the 3d Courant Sigma Model associated with the double of the underlying Lie bialgebroid. This field-theoretic result follows from a Lie-theoretic one involving a coisotropic reduction of the odd cotangent bundle by a generalized space of algebroid paths. We also provide several examples, including the case of symplectic groupoids in which we relate the symplectic realization construction of Crainic–Marcut to a particular gauge fixing of the 3d theory.

REDUCTION OF SYMPLECTIC GROUPOIDS AND QUOTIENTS OF QUASI-POISSON MANIFOLDS

In this work, we study the integrability of quotients of quasi-Poisson manifolds. Our approach allows us to put several classical results about the integrability of Poisson quotients in a common framework. By categorifying one of the already known methods of reducing symplectic groupoids we also describe double symplectic groupoids, which integrate the recently introduced Poisson groupoid structures on gauge groupoids.

Linearization of Poisson groupoids

Motivated by a search for Lie group structures on groups of Poisson diffeomorphisms, 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 bialgebroid to a symplectic LA-groupoid, and in the process we define interesting new examples of double Lie algebroids and LA-groupoids. We also show that the product Poisson groupoid can only be linearizable when the Poisson structure on the unit space is regular.

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Given a standard complex semisimple Poisson Lie group $(G, \pi_{st})$, generalised double Bruhat cells $G^{u, v}$ and generalised Bruhat cells $O^u$ equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang Hua Lu and the author. We prove in this paper that $G^{u,u}$ is naturally a Poisson groupoid over $O^u$, extending a result from the aforementioned authors about double Bruhat cells in $(G, \pi_{st})$. Our result on $G^{u,u}$ is obtained as an application of a construction interesting in its own right, of a local Poisson groupoid over a mixed product Poisson structure associated to the action of a pair of Lie bialgebras. This construction involves using a local Lagrangian bisection in a double symplectic groupoid closely related to the global R-matrix studied by Weinstein and Xu, to twist a direct product of Poisson groupoids.

Variational order for forced Lagrangian systems II. Euler–Poincaré equations with forcing

In this paper we provide a variational derivation of the Euler–Poincaré equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others Martín de Diego and Martín de Almagro (2018 Nonlinearity 31 3814–3846), Galley (2013 Phys. Rev. Lett. 110 174301), Galley et al (2014 (arXiv:[math-Ph] 1412.3082)). Moreover, we study in detail the underlying geometry which is related to the notion of Poisson groupoid. Finally, we apply the previous construction to the formal derivation of the variational error for numerical integrators of forced Euler–Poincaré equations and the application of this theory to the derivation of geometric integrators for forced systems.

Poisson Geometry Related to Atiyah Sequences

We construct and investigate a short exact sequence of Poisson $\mathcal{VB}$-groupoids which is canonically related to the Atiyah sequence of a $G$-principal bundle $P$. Our results include a description of the structure of the symplectic leaves of the Poisson groupoid $\frac{T^*P\times T^*P}{G}\rightrightarrows \frac{T^*P}{G}$. The semidirect product case, which is important for applications in Hamiltonian mechanics, is also discussed.

On a Poisson homogeneous space of bilinear forms with a Poisson–Lie action

Let be the space of bilinear forms on with defining matrices endowed with a quadratic Poisson structure of reflection equation type. The paper begins with a short description of previous studies of the structure, and then this structure is extended to systems of bilinear forms whose dynamics is governed by the natural action of the Poisson–Lie group on . A classification is given of all possible quadratic brackets on preserving the Poisson property of the action, thus endowing with the structure of a Poisson homogeneous space. Besides the product Poisson structure on , there are two other (mutually dual) structures, which (unlike the product Poisson structure) admit reductions by the Dirac procedure to a space of bilinear forms with block upper triangular defining matrices. Further generalisations of this construction are considered, to triples with the Poisson action , and it is shown that then acquires the structure of a Poisson symmetric space. Generalisations to chains of transformations and to the quantum and quantum affine algebras are investigated, as well as the relations between constructions of Poisson symmetric spaces and the Poisson groupoid. Bibliography: 30 titles.

DOUBLE BRUHAT CELLS AND SYMPLECTIC GROUPOIDS

Let G be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure π st determined by a pair of opposite Borel subgroups (B, B_). We prove that for each υ in the Weyl group W of G, the double Bruhat cell G υ,υ = BυB Ω B_υB_ in G, together with the Poisson structure π st, is naturally a Poisson groupoid over the Bruhat cell BυB/B in the flag variety G/B. Correspondingly, every symplectic leaf of π st in G υ,υ is a symplectic groupoid over BυB/B. For u, υ ϵ W, we show that the double Bruhat cell (G u,υ , π st) has a naturally defined left Poisson action by the Poisson groupoid (G u,υ , π st) and a right Poisson action by the Poisson groupoid (G u,υ , π st), and the two actions commute. Restricting to symplectic leaves of π st, one obtains commuting left and right Poisson actions on symplectic leaves in G u,υ by symplectic leaves in G u,u and G υ,υ as symplectic groupoids.

Almost regular Poisson manifolds and their holonomy groupoids

We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian vector fields arises from a smooth holonomy groupoid. We call them almost regular Poisson structures and determine them completely. They include regular Poisson and log symplectic manifolds, as well as several other Poisson structures whose symplectic foliation presents singularities. We show that the holonomy groupoid associated with an almost regular Poisson structure is a Poisson groupoid, integrating a naturally associated Lie bialgebroid. The Poisson structure on the holonomy groupoid is regular, and as such it provides a desingularization. The holonomy groupoid is "minimal" among Lie groupoids which give rise to the submodule generated by Hamiltonian vector fields. This implies that, in the case of log-symplectic manifolds, the holonomy groupoid coincides with the symplectic groupoid constructed by Gualtieri and Li. Last, we focus on the integrability of almost regular Poisson manifolds and exhibit the role of the second homotopy group of the source-fibers of the holonomy groupoid.

Homogeneous spaces of Dirac groupoids

A Poisson structure on a homogeneous space of a Poisson groupoid is homogeneous if the action of the Lie groupoid on the homogeneous space is compatible with the Poisson structures. According to a result of Liu, Weinstein and Xu, Poisson homogeneous spaces of a Poisson groupoid are in correspondence with suitable Dirac structures in the Courant algebroid defined by the Lie bialgebroid of the Poisson groupoid. We show that this correspondence result fits into a more natural context: the one of Dirac groupoids , which are objects generalizing Poisson groupoids and multiplicative closed 2-forms on groupoids.

Generalized Classical Dynamical Yang-Baxter Equations and Moduli Spaces of Flat Connections on Surfaces

In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new finite dimensional description of the Atiyah-Bott symplectic structure on the moduli space of flat connections on a surface.

Study of Combinatorial Operator on Poisson Manifolds

Generalized Hamilton system is defined by generalized Poisson bracket, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion,. In the paper, Combinatorial operator in 1-form space on Poisson manifold P is constructed and sufficient and necessary conditions that the vector field which induced by 1-form is symplectic vector field are offered, that is, Poisson bracket is a special type of combinatorial operator. On the basis of the above discussion, some properties about the combinatorial operator in Poisson manifold and groupoid are obtained. All the -invariant 1-form induced by Poisson structure construct the Lie subalgebra of .

Poisson Involutions, Spin Calogero-Moser Systems Associated with Symmetric Lie Subalgebras and the Symmetric Space Spin Ruijsenaars-Schneider Models

We develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary Poisson groupoids. The method is based on the successive use of Dirac reduction and Poisson reduction. Then we show that certain spin Calogero-Moser systems associated with symmetric Lie subalgebras can be studied in this fashion. We also consider some spin-generalized Ruisjenaars- Schneider equations which correspond to the $N$-soliton solutions of $A^{(1)}_n$ affine Toda field theory. In this case, we show how the equations are obtained from the Dirac reduction of some Hamiltonian system on a symmetric coboundary dynamical Poisson groupoid.

On transitive Lie bialgebroids and Poisson groupoids

We prove that, for any transitive Lie bialgebroid ( A , A ∗ ), the differential associated to the Lie algebroid structure on A ∗ has the form d ∗ = [ Λ , ⋅ ] A + Ω , where Λ is a section of ∧ 2 A and Ω is a Lie algebroid 1-cocycle for the adjoint representation of A . Globally, for any transitive Poisson groupoid ( Γ , Π ) , the Poisson structure has the form Π = Λ ← − Λ → + Π F , where Π F is a bivector field on Γ associated to a Lie groupoid 1-cocycle.

Quasi-bialgebras and dynamical [formula omitted]-matrices

We study the relationship between general dynamical Poisson groupoids and Lie quasi-bialgebras. For a class of Lie quasi-bialgebras G naturally compatible with a reductive decomposition, we extend the description of the moduli space of classical dynamical r-matrices of Etingof and Schiffmann. We construct, in each gauge orbit, an explicit analytic representative l can . We translate the notion of duality for dynamical Poisson groupoids into a duality for Lie quasi-bialgebras. It is shown that duality maps the dynamical Poisson groupoid for l can and G to the dynamical Poisson groupoid for l can and the dual quasi-bialgebra G ★ .

Dirac submanifolds and Poisson involutions

Dirac submanifolds are a natural generalization in the Poisson category of the symplectic submanifolds of a symplectic manifold. They correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable loci of Poisson involutions. In this paper, we make a general study of these submanifolds including both local and global aspects. In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable loci. In particular, we discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups, and symmetric Poisson groupoids. Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoid structures arising from dynamical r-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we prove that the Dubrovin Poisson structure on the space of Stokes matrices U + (appearing in Dubrovin's theory of Frobenius manifolds) is a Poisson symmetric space.

Integration of Lie bialgebroids

We prove that under certain mild assumptions a Lie bialgebroid integrates to a Poisson groupoid. This includes, in particular, a new proof of the existence of local symplectic groupoids for any Poisson manifold, a theorem of Karasev and of Weinstein.

Classical lifting processes and multiplicative vector fields

We extend the calculus of multiplicative vector fields and differential forms and their intrinsic derivatives from Lie groups to Lie groupoids; this generalization turns out to include also the classical process of complete lifting from arbitrary manifolds to tangent and cotangent bundles. Using this calculus we give a new description of the Lie bialgebroid structure associated with a Poisson groupoid.

Classical Lifting Processes and Multiplicative Vector Fields

We extend the calculus of multiplicative vector fields and differential forms and their intrinsic derivatives from Lie groups to Lie groupoids; this generalization turns out to include also the classical process of complete lifting from arbitrary manifolds to tangent and cotangent bundles. Using this calculus we give a new description of the Lie bialgebroid structure associated with a Poisson groupoid.