A family of Poisson structures, parametrised by an arbitrary odd periodic function $ϕ$, is defined on the space $\cW$ of twisted polygons in $\RR^ν$. Poisson reductions with respect to two Poisson group actions on $\cW$ are described. The $ν=2$ and $ν=3$ cases are discussed in detail and the general $ν$ case in less detail. Amongst the Poisson structures arising in examples are to be found the lattice Virasoro structure, the second Toda lattice structure and some extended Toda lattice structures. A general result is proved showing that, for any $ν$, to certain concrete choices of $ϕ$ there correspond compatible Poisson structures which generate all the extended bigraded Toda hierarchies of a suitable size.
The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, study the corresp…
We discuss dimensional reduction for Hamiltonian systems which possess nonconstant Poisson brackets between pairs of coordinates and between pairs of momenta. The associated Jacobi identities imply that the dimensionally reduced brackets are always constant. Some examples are given alongside the general theory.
Symmetrical top is a special case of a general top. The canonical Poisson structure on T*SE(3) is the common method of its description. This Poisson structure is invariant under the right action of SO(3). However the Hamiltonian of the symmetrical top is only invariant under the right action of the subgroup S1 that corresponds to the rotation around the top symmetry axis. Then the Poisson structure for a symmetric top is obtained by reduction T*SE(3)/S1 is shown.
We study the invariants (in particular, the central invariants) of suitable Poisson pencils from the point of view of the theory of bi-Hamiltonian reduction, paying a particular attention to the case where the Poisson pencil is exact. We show that the exactness is preserved by the reduction. In the Drinfeld-Sokolov case, the same is true for the characteristic polynomial of the pencil, which plays a crucial role in the definition of the central invariants. We also discuss the bi-Hamiltonian structures of a generalized Drinfeld-Sokolov hierarchy and of the Camassa-Holm equation.
We show that the 2d Poisson Sigma Model on a Poisson groupoid arises as an effective theory of the 3d Courant Sigma Model associated to 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.
A new notion of a dual Poisson-presymplectic pair is introduced and its properties are examined. The procedure of Dirac reduction of Poisson operators onto submanifolds proposed by Dirac is in this paper embedded in a geometric procedure of reduction of dual Poisson-presymplectic pairs. The method presented generalizes those used by Marsden and Ratiu for reductions of Poisson manifolds. Two examples are given.
The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by Ib$\acute{\text{a}}\tilde{\text{n}}$ez et al. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduction theorem for Nambu-Poisson manifolds. Finally, we define gauge transformations of Nambu-Poisson structures and show that these transformations commute with the reduction procedure.
This paper develops the theory of affine Lie-Poisson reduction and applies this process to Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids. As a consequence of this approach, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples.
The Dirac reduction technique used previously to obtain solutions of the classical dynamical Yang-Baxter equation on the dual of a Lie algebra is extended to the Poisson-Lie case and is shown to yield naturally certain dynamical r-matrices on the duals of Poisson-Lie groups found by Etingof, Enriquez and Marshall in math.QA/0403283.
We propose a generalization of the reduction of Poisson manifolds by distributions introduced by Marsden and Ratiu. Our proposal overcomes some of the restrictions of the original procedure, and makes the reduced Poisson structure effectively dependent on the distribution. Different applications are discussed, as well as the algebraic interpretation of the procedure and its formulation in terms of Dirac structures.
We prove the existence and stability of flat steady states of the Vlasov-Poisson system, which in astrophysics are used as models of disk-like galaxies. We follow the variational approach developed by Guo and Rein for this type of problems and extend previous results of Rein. In particular, we employ a reduction procedure which relates the stability problem for the Vlasov-Poisson system to the analogous question for the Euler-Poisson system.
This paper studies the reduced dynamics of the three-vortex problem from the point of view of Lie-Poisson reduction on the dual of the Lie algebra of $ U(2) $. The algebraic study leading to this point of view has been given by Borisov and Lebedev 1998 (see also Bolsinov, Borisov and Mamaev 1999). The main contribution of this paper is to properly describe the dynamics as a Lie-Poisson reduced system on $ (\mathfrak{u}(2) ^\ast, \{\;,\,\}_{\text{LP}} ) $, giving a systematic construction of a one-parameter family of covectors $ \{ σ_1, σ_2, σ_3 \} $ closely related to Pauli spin matrices, and …
BRST complexes are differential graded Poisson algebras. They are associated to a coisotropic ideal $J$ of a Poisson algebra $P$ and provide a description of the Poisson algebra $(P/J)^J$ as their cohomology in degree zero. Using the notion of stable equivalence introduced by Felder and Kazhdan, we prove that any two BRST complexes associated to the same coisotropic ideal are quasi-isomorphic in the case $P = \mathbb{R}[V]$ where $V$ is a finite-dimensional symplectic vector space and the bracket on $P$ is induced by the symplectic structure on $V$. As a corollary, the cohomology of the BRST c…
We construct an analogue of Dirac's reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac's reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld-Sokolov hierarchy.
We consider the Poisson reduced space $(T^*Q)/K$ with respect to a cotangent lifted action. It is assumed that $K$ is a compact Lie group which acts by isometries on the Riemannian manifold $Q$ and that the action on $Q$ is of single isotropy type. Realizing $(T^*Q)/K$ as a Weinstein space we determine the induced Poisson structure and its symplectic leaves. We thus extend the Weinstein construction for principal fiber bundles to the case of surjective Riemannian submersions $Q\to Q/K$.
We show how to reduce, under certain regularities conditions, a Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson-Nijenhuis manifolds. The choice of the more general framework of Lie algebroids is motivated by the geometrical study of some reduced bi-Hamiltonian systems. An explicit example of reduction of a Poisson-Nijenhuis Lie algebroid is also provided.
Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem -- connecting symmetries with constants of the motion -- within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic formalism.
We give a definition of coisotropic morphisms of shifted Poisson (i.e. $P_n$) algebras which is a derived version of the classical notion of coisotropic submanifolds. Using this we prove that an intersection of coisotropic morphisms of shifted Poisson algebras carries a Poisson structure of shift one less. Using an interpretation of Hamiltonian spaces as coisotropic morphisms we show that the classical BRST complex computing derived Poisson reduction coincides with the complex computing coisotropic intersection. Moreover, this picture admits a quantum version using brace algebras and their mod…
The trigonometric Ruijsenaars-Schneider model is derived by symplectic reduction of Poisson-Lie symmetric free motion on the group U(n). The commuting flows of the model are effortlessly obtained by reducing canonical free flows on the Heisenberg double of U(n). The free flows are associated with a very simple Lax matrix, which is shown to yield the Ruijsenaars-Schneider Lax matrix upon reduction.
We revisit our earlier work on the AKSZ formulation of topological sigma model on generalized complex manifolds, or Hitchin model. We show that the target space geometry geometry implied by the BV master equations is Poisson--quasi--Nijenhuis geometry recently introduced and studied by Stiénon and Xu (in the untwisted case). Poisson--quasi--Nijenhuis geometry is more general than generalized complex geometry and comprises it as a particular case. Next, we show how gauging and reduction can be implemented in the Hitchin model. We find that the geometry resulting form the BV master equation is c…
We construct recursively an infinite number of Poisson structures for the supersymmetric integrable hierarchy governing the Pohlmeyer reduction of superstring sigma models on the target spaces AdS_{n}\times S^n, n=2,3,5. These Poisson structures are all non-local and not relativistic except one, which is the canonical Poisson structure of the semi-symmetric space sine-Gordon model (SSSSG). We verify that the superposition of the first three Poisson structures corresponds to the canonical Poisson structure of the reduced sigma model. Using the recursion relations we construct commuting charges …
The classical Poisson reduction of a given Lagrangian system with (local) gauge symmetries has to be done before its quantization. We propose here a coordinate free and self-contained mathematical presentation of the covariant Batalin-Vilkovisky Poisson reduction of a general gauge theory. It was explained in physical terms (DeWitt indices) in Henneaux and Teitelboim's book on quantization of gauge theories. It was studied in coordinates using jet spaces by Barnich-Brandt-Henneaux and Stasheff, among others. The main idea of our approach is to use the functor of point approach to spaces of fie…
We derive a Hamiltonian structure for the $N$-particle hyperbolic spin Ruijsenaars-Schneider model by means of Poisson reduction of a suitable initial phase space. This phase space is realised as the direct product of the Heisenberg double of a factorisable Lie group with another symplectic manifold that is a certain deformation of the standard canonical relations for $N\ell$ conjugate pairs of dynamical variables. We show that the model enjoys the Poisson-Lie symmetry of the spin group ${\rm GL}_{\ell}({\mathbb C})$ which explains its superintegrability. Our results are obtained in the formal…
This manuscript is essentially a collection of lecture notes which were given by the first author at the Summer School Wisl-2019, Poland and written down by the second author. As the title suggests, the material covered here includes the Poisson and symplectic structures (Poisson manifolds, Poisson bi-vectors and Poisson brackets), group actions and orbits (infinitesimal action, stabilizers and adjoint representations), moment maps, Poisson and Hamiltonian actions. Finally, the phase space reduction is also discussed. The very last section introduces the Poisson-Lie structures along with some …
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 obtai…
In this paper we propose a reduction scheme for multivector fields phrased in terms of $L_\infty$-morphisms. Using well-know geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to built up a suitable deformation retract of DGLA's. We first obtained an explicit formula for the $L_\infty$-Projection and -Inclusion of generic DGLA retracts. We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction $L_\infty$-morphis…
We show that the symplectic reduction of the dynamics of $N$ point vortices on the plane by the special Euclidean group $\mathsf{SE}(2)$ yields a Lie--Poisson equation for relative configurations of the vortices. Specifically, we combine symplectic reduction by stages with a dual pair associated with the reduction by rotations to show that the $\mathsf{SE}(2)$-reduced space with non-zero angular impulse is a coadjoint orbit. This result complements some existing works by establishing a relationship between the symplectic/Hamiltonian structures of the original and reduced dynamics. We also find…
The nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demonstrate the integrability of the equations for some especially important partial classes of compatible nonlocal Poisson brackets of hydrodynamic type, in particular, for the most important case when one of the compatible Poisson brackets is local and also for the case when one of…
We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply our method to derive a classical $r$-matrix for the elliptic Calogero-Moser system with spin starting from the Higgs bundle over an elliptic curve with marked points. In the paper we also derive a clas…
We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group ${\mathrm{GL}}(n,{\mathbb{C}})$, and study its Poisson reduction with respect to the action of the product group ${{\mathrm U}(n)} \times {{\mathrm U}(n)}$ arising from left- and right-multiplications. One of the pertinent Poisson structures is the canonical one, while the other is suitably transferred from the real Heisenberg double of ${\mathrm{GL}}(n,{\mathbb{C}})$. When taking the quotient of $T^*{\mathrm{GL}}(n,{\mathbb{C}})$ we focus on the dense open subset of ${\mathrm{GL}}(n,{\mathbb{C}})$ whose elem…
A new procedure for the global construction of the Casimir invariants and Darboux canonical form for finite-dimensional Poisson systems is developed. This approach is based on the concept of matrix congruence and can be applied without the previous determination of the Casimir invariants (recall that their prior knowledge is unavoidable for the standard reduction methods, thus requiring either the integration of a system of PDEs or solving some equivalent problem). Well the opposite, in the new congruence method, both the Darboux coordinates and the Casimir invariants arise simultaneously as t…
We first exhibit two compatible Poisson structures on the cotangent bundle of the unitary group $\mathrm{U}(n)$ in such a way that the invariant functions of the $\mathfrak{u}(n)^*$-valued momenta generate a bi-Hamiltonian hierarchy. One of the Poisson structures is the canonical one and the other one arises from embedding the Heisenberg double of the Poisson-Lie group $\mathrm{U}(n)$ into $T^*\mathrm{U}(n)$, and subsequently extending the embedded Poisson structure to the full cotangent bundle. We then apply Poisson reduction to the bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$ using the con…
We consider nonholonomic systems which symmetry groups consist of two subgroups one of which represents rotations about the axis of symmetry. After nonholonomic reduction by another subgroup the corresponding vector fields on partially reduced phase space are linear combinations of the Hamiltonian and symmetry vector fields. The reduction of the Poisson bivectors associated with the Hamiltonian vector fields to canonical form is discussed.
In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to use very general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective…
Symplectic reduction is reinterpreted as the composition of arrows in the category of integrable Poisson manifolds, whose arrows are isomorphism classes of dual pairs, with symplectic groupoids as units. Morita equivalence of Poisson manifolds amounts to isomorphism of objects in this category. This description paves the way for the quantization of the classical reduction procedure, which is based on the formal analogy between dual pairs of Poisson manifolds and Hilbert bimodules over C*-algebras, as well as with correspondences between von Neumann algebras. Further analogies are drawn with ca…
Extending our reduction construction in \cite{Hu} to the Hamiltonian action of a Poisson Lie group, we show that generalized Kähler reduction exists even when only one generalized complex structure in the pair is preserved by the group action. We show that the constructions in string theory of the (geometrical) $T$-duality with $H$-fluxes for principle bundles naturally arise as reductions of factorizable Poisson Lie group actions. In particular, the group may be non-abelian.
It is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic $r$-matrix Poisson brackets on the associative algebra ${\mathrm{gl}}(n,{\mathbb{R}})$. We here show that the quadratic bracket on ${\mathrm{gl}}(n,{\mathbb{R}})$, corresponding to the $r$-matrix defined by the splitting of ${\mathrm{gl}}(n,{\mathbb{R}})$ into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle $T…
We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in SU(2), denoted $C_r^n$, by the diagonal conjugation action of SU(2). Here $C_r^n$ is a quasi-Hamiltonian SU(2)-space. An integrable Hamiltonian system is constructed on $M_r$ in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on $…
In his pioneering paper [Phys. Rev. E 7, 2405 (1973)], Nambu proposed the idea of multiple Hamiltonian systems. The explicit example examined there is equivalent to the so(3) Lie-Poisson system, which represents noncanonical Hamiltonian dynamics with a Casimir; the Casimir corresponds to the second Hamiltonian of Nambu's formulation. The vortex dynamics of ideal fluid, while it is infinite dimensional, has a similar structure, in which the Casimir is the helicity. These noncanonical Poisson algebras are derived by the reduction, i.e., restricting the phase space to some submanifold embedded in…
