bi-Poisson Structures Research Articles

Reductions of Invariant bi-Poisson Structures and Locally Free Actions

Let (X,G,ω1,ω2,{ηt}) be a manifold with a bi-Poisson structure {ηt} generated by a pair of G-invariant symplectic structures ω1 and ω2, where a Lie group G acts properly on X. We prove that there exists two canonically defined manifolds (RLi,Gi,ω1i,ω2i,{ηit}), i=1,2 such that (1) RLi is a submanifold of an open dense subset X(H)⊂X; (2) symplectic structures ω1i and ω2i, generating a bi-Poisson structure {ηit}, are Gi- invariant and coincide with restrictions ω1|RLi and ω2|RLi; (3) the canonically defined group Gi acts properly and locally freely on RLi; (4) orbit spaces X(H)/G and RLi/Gi are canonically diffeomorphic smooth manifolds; (5) spaces of G-invariant functions on X(H) and Gi-invariant functions on RLi are isomorphic as Poisson algebras with the bi-Poisson structures {ηt} and {ηit} respectively. The second Poisson algebra of functions can be treated as the reduction of the first one with respect to a locally free action of a symmetry group.

Transverse hilbert schemes, bi-hamiltonian systems and hyperkähler geometry

Abstract We give a characterization of Atiyah’s and Hitchin’s transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces.

Classical Affine $${\mathcal{W}}$$-Algebras for $${\mathfrak{gl}_N}$$ and Associated Integrable Hamiltonian Hierarchies

We apply the new method for constructing integrable Hamiltonian hierarchies of Lax type equations developed in our previous paper, to show that all W-algebras W(gl_N,f) carry such a hierarchy. As an application, we show that all vector constrained KP hierarchies and their matrix generalizations are obtained from these hierarchies by Dirac reduction, which provides the former with a bi-Poisson structure.

Two-component integrable generalizations of Burgers equations with nondiagonal linearity

Two-component second- and third-order Burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries. New integrable systems are obtained. Master symmetries of the obtained symmetry integrable systems, and bi-Poisson structures of those that also possess conservation laws, are given.

Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras

We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th KdV hierarchy, using the formal distribution calculus and the lambda-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (non-local) bi-Poisson structures of the matrix KP and the matrix N-th KdV hierarchies, and we prove integrability of the N-th matrix KdV hierarchy.

Poisson structure of a modified Hunter–Saxton equation

Both the well-known Korteweg–de Vries equation and the Hunter–Saxton equation with a linear wux-term, −utxx = −2ωux + 2uxuxx + uuxxx, ω > 0, are bi-Hamiltonian in the sense that each equation is Hamiltonian with respect to two compatible, but distinct, Poisson brackets. We present a nonlinear change of variables which maps between the two bi-Poisson structures, giving a correspondence between the infinite hierarchies of conservation laws and commuting flows associated with the equations. In particular, under this change of variables this (modified) Hunter–Saxton equation can be viewed as a member of the KdV hierarchy.

Systems of Hess-Appel’rot Type

We construct higher-dimensional generalizations of the classical Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter leading to an algebro-geometric integration of this new class of systems, which is closely related to the integration of the Lagrange bitop performed by us recently and uses Mumford relation for theta divisors of double unramified coverings. Based on the basic properties satisfied by such a class of systems related to bi-Poisson structure, quasi-homogeneity, and conditions on the Kowalevski exponents, we suggest an axiomatic approach leading to what we call the "class of systems of Hess-Appel'rot type".

COMPATIBLE POISSON STRUCTURES OF TODA TYPE DISCRETE HIERARCHY

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and nonlocal families of R-matrix solutions to the modified Yang–Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.

Bi-Poisson Structures and Integrability of Geodesic Flow on Homogeneous Spaces

Let G=K be a semisimple orbit of the adjoint representation of a real connected reductive Lie group G. Let K1 be any closed subgroup of K containing the commutant of the identity component of K. We prove that the geodesic flow on the symplectic manifold T ⁄ (G=K1), corresponding to a G-invariant pseudo-Riemannian metric on G=K1 which is in- duced by a bi-invariant pseudo-Riemannian metric on G, is completely integrable in the class of real analytic functions, polynomial in momenta. To this end we study the Poisson geometry of the space of G-invariant functions on T ⁄ (G=K) using a one-parameter family of moment maps.

Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems

We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction (performed with respect to a special type action of a Lie group) of micro-Jordan bihamiltonian structures whose Nijenhuis tensor has constant eigenvalues. The method is applied to the diagonal action of a Lie group G on a direct product of N coadjoint orbits O=O 1×⋯×O N⊂ g ∗×⋯× g ∗ endowed with a bihamiltonian structure whose first generator is the standard symplectic form on O . As a result we get the so-called classical Gaudin system on O . The method works for a wide class of Lie algebras including the semisimple ones and for a large class of orbits including the generic ones and the semisimple ones.