Flat pencils of symplectic connections and Hamiltonian operators of degree 2

Abstract

Bi-Hamiltonian structures involving Hamiltonian operators of degree 2 are studied. Firstly, pairs of degree 2 operators are considered in terms of an algebra structure on the space of 1-forms, related to so-called Fermionic Novikov algebras. Then, degree 2 operators are considered as deformations of hydrodynamic type Poisson brackets.

Projective-geometric aspects of homogeneous third-order Hamiltonian operators

“…Higher-order operators were subsequently defined in [7]. The structure of homogeneous second-order Hamiltonian operators was investigated in [21,5], see also [14].…”

Geometry of inhomogeneous Poisson brackets, multicomponent Harry Dym hierarchies and multicomponent Hunter-Saxton equations

“…For even k non-degenerate skew-symmetric tensor h αβ is actually the inverse of symplectic structure and Γ β rs is flat symplectic connection. The corresponding pair defines the so called structure of flat Fedosov manifold [17], [18], which is the main ingredient of Fedosov quantization procedure. To study the analogs of Harry Dym equations in this case is an interesting problem, but it is out of the scope of this paper.…”

Second-Order Deformations of Hydrodynamic-Type Poisson Brackets

“…a flat symplectic connection (see, for example, [2]). In [7] such a pair (ω, ∇) was referred to as a Fedosov structure.…”

“…Section 3 considers the properties of inhomogeneous Hamiltonian operators containing degree 1 and degree 2 parts. We refer the interested reader to [7] for further details and proofs.…”