We construct and classify all Poisson structures on quasimodular forms that extend the one coming from the first Rankin-Cohen bracket on the modular forms. We use them to build formal deformations on the algebra of quasimodular forms.

It is shown that two canonical maps arising in the Poisson bracket formulations of elasticity and superfluids are particular instances of general canonical maps between duals of semidirect product Lie algebras.

Given a Poisson structure (or, equivalently, a Hamiltonian operator) P , we show that its Lie derivative Lτ (P ) along a vector field τ defines another Poisson structure, which is automatically compatible with P , if and only if [L τ (P ), P ] = 0, where [·, ·] is the Schouten bracket. This result yields a new local description for the set of all Poisson structures compatible with a given Poiss...

The Clifford algebraic formulation of the Duffin–Kemmer–Petiau (DKP) algebras is applied to recast De Donder–Weyl Hamiltonian (DWH) theory as an description independent matrix representation DKP algebra. We show that DWH equations for antisymmetric fields arise out action algebra on certain invariant subspaces which carry representations fields. representation-free formula bracket associated wi...

We extend the Kontsevich formality L∞-morphism U : T • poly(R ) → D poly(R ) to an L∞-morphism of an L∞-modules over T • poly(R ), Û : C•(A, A) → Ω(R), A = C(R). The construction of the map Û is given in Kontsevich-type integrals. The conjecture that such an L∞-morphism exists is due to Boris Tsygan [Ts]. As an application, we obtain an explicit formula for isomorphism A∗/[A∗, A∗] ∼ → A/{A, A} ...

The close interplay between geometry, topology and algebra turned out to be a most crucial point in the analysis of low dimensional field theories. This is particularly true for the large class of topological and almost topological two-dimensional field theories which can be treated comprehensively in the framework of Poisson-σ models. Examples are provided by pure YangMills and gravity theorie...

Dynamical systems, finite or infinite, that describe physical phenomena typically have parts that are in some sense Hamiltonian and parts that can be recognized as dissipative, with the Hamiltonian part being generated by a Poisson bracket and the dissipative part being some kind of gradient flow. The description of Hamiltonian systems has received much attention over nearly two centuries and, ...

Abstract. We study the Poisson structure associated to the defocusing Ablowitz-Ladik equation from a functional-analytical point of view, by reexpressing the Poisson bracket in terms of the associated Carathéodory function. Using this expression, we are able to introduce a family of compatible Poisson brackets which form a multi-Hamiltonian structure for the Ablowitz-Ladik equation. Furthermore...

In a companion paper, we introduced a notion of multi-Dirac structures, a graded version of Dirac structures, and we discussed their relevance for classical field theories. In the current paper we focus on the geometry of multi-Dirac structures. After recalling the basic definitions, we introduce a gradedmultiplication and amulti-Courant bracket on the space of sections of a multi-Dirac structu...

The structure of Poisson polynomial algebras of the type obtained as semiclas-sical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equi...