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Crawley-Boevey introduced in [Poisson structure on moduli spaces of representations, J. Algebra 325 (2011) 205–215.], the notion of [Formula: see text]-Poisson structure for associative algebras, which is the weakest condition that induces a Poisson structure on the moduli spaces of their representations. In this paper, by using a result of Armenta and Keller in [Derived invariance of the Tamarkin-Tsygan calculus of an algebra, C. R. Math. Acad. Sci. Paris 357(3) (2019) 236–240.], we show that an [Formula: see text]-Poisson structure is preserved under derived Morita equivalence.

We construct certain new classes of isochronous dynamical systems based on the recent constructions of Calogero and Leyvraz. We show how a Poisson structure can be ascribed to such equations in ℝ3 and indicate their connection with the Nambu structures.

We prove that when Kontsevich's deformation quantization is applied on weight homogeneous Poisson structures, the operators in the $\ast-$ product formula are weight homogeneous. We then consider the linear Poisson case $X=\mathfrak{g}^\ast$ for a semi simple Lie algebra $\mathfrak{g}$. As an application we provide an isomorphism between the Cattaneo-Felder-Torossian reduction algebra $H^0(\mathfrak{g},\mathfrak{m},\chi)$ and the $W-$ algebra $(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{m}_\chi)^\mathfrak{m}$. We also show that in the $W-$ algebra setting, $(S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{m}_\chi)^\mathfrak{m}$ is polynomial. Finally, we compute generators of $H^0(\mathfrak{g},\mathfrak{m},\chi)$ as a deformation of $(S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{m}_\chi)^\mathfrak{m}$., Comment: expanded version, 19 pages, 3 figures

We start with a review of a class of systems with invariant relations, so called {\it systems of Hess--Appel'rot type} that generalizes the classical Hess--Appel'rot rigid body case. The systems of Hess-Appel'rot type carry an interesting combination of both integrable and non-integrable properties. Further, following integrable line, we study partial reductions and systems having what we call the {\it Zhukovskii property}: these are Hamiltonian systems with invariant relations, such that partially reduced systems are completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess-Appel'rote type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess-Appel'rot type - the non-integrable part, some analysis of which may be seen as a reconstruction problem. We show that an integrable system, the magnetic pendulum on the oriented Grassmannian $Gr^+(4,2)$ has natural interpretation within Zhukovskii property and it is equivalent to a partial reduction of certain system of Hess-Appel'rot type. We perform a classical and an algebro-geometric integration of the system, as an example of an isoholomorphic system. The paper presents a lot of examples of systems of Hess-Appel'rot type, giving an additional argument in favor of further study of this class of systems., Comment: 42 pages

Deformation quantization of Poisson manifolds is studied within the framework of an expansion in powers of derivatives of Poisson structures. We construct the Lie group associated with a Poisson bracket algebra which defines a second order deformation in the derivative expansion., 7 pages,minor corrections; v3: details added

We discuss some geometrical and dynamical properties of Rabinovich systems.

Recently Nieto has proposed a link between oriented matroid theory and the Schild type action of p-branes. This particular matroid theory satisfies the local condition, i.e., the degenerate form must be closed. This allows us to explain the dynamics of p-branes in terms of Nambu–Poisson structure. In this paper using an infinitesimal canonical transformation of Nambu brackets we show that the helicity is conserved in the dynamics of p-branes. Applying Filippov algebra (or quantum Nambu bracket) we define a generalized Yang–Mills action in 4k space. We show that this action is equivalent to Dolan–Tchrakian type action.

The concept of a canonical Bäcklund transformation as laid down by Sklyanin is extended to a discrete integral chain, with a Poisson structure which is not canonical in the strict sense. The transformation is induced by an auxiliary Lax operator with a classical r-matrix which is similar in its algebraic structure to that of the original Lax operator governing the dynamics of the chain. Moreover, the transformation can be obtained from a suitable generating function. It is also shown how successive transformations can be composed to construct a new transformation. Finally an inverse transformation is also constructed. The compatibility of the transformation with the "time" part of the Lax equation is explicitly demonstrated. It is also shown that the Bianchi theorem of permutability holds good.

The Poisson structure related to [Formula: see text]-Poincaré Group is explicitly presented as a dual to a certain Lie algebroid in a geometric and coordinate-free way. The related Poisson structure on the (affine) Minkowski space is also presented in a geometric way.

We relate a universal formula for the deformation quantization of Poisson structures (⋆-products) on ℝd proposed by Maxim Kontsevich to the Campbell–Baker–Hausdorff (CBH) formula. We show that Kontsevich's formula can be viewed as exp (P) where P is a bi-differential operator that is a deformation of the given Poisson structure. For linear Poisson structures (duals of Lie algebras) his product takes the form exp (C+L) where exp (C) is the ⋆-product given by the universal enveloping algebra via symmetrization, essentially the CBH formula. This is established by showing that the two products are identical on duals of nilpotent Lie algebras where the operator L vanishes. This completely determines part of Kontsevich's formula and leads to a new scheme for computing the CBH formula. The main tool is a graphical analysis for bi-differential operators and the computation of certain iterated integrals that yield the Bernoulli numbers.

+ {g1, {f1, . . . , fn−1, g2}, g3, . . . , gn} + · · ·+ {g1, . . . , gn−1, {f1, . . . , fn−1, gn}} for all f1, . . . , fn−1, g1, . . . , gn ∈ C∞(M). We should note that (f1, . . . , fn−1) acts on {g1, . . . , gn} as a derivation. If n = 2, we have usual Poisson manifolds. But if n ≥ 3, there appear some aspects which are different from the case of usual Poisson manifolds. More precisely, Nambu–Poisson structure should be more rigid than usual Poisson structure. (For example, see Theorem 5.5.) P. Gautheron [3] also proved the same result as ours in a completely different way. Using the Fundamental Identity, we know that the flow of the equation of motion induces an automorphism of a Nambu–Poisson bracket.

In this talk we describe the Poisson structure of the moduli space of flat connections on a two dimensional Riemann surface in terms of lattice gauge fields and Poisson–Lie groups.

A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an 'integral' of the Koszul–Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dμ, dν]P=d{μ, ν}P. A naturally defined generalized Hamiltonian map is proved to be a homomorphism between { , }P and the Frölicher–Nijenhuis bracket of vector valued forms. Also relations of this graded Poisson bracket to the Schouten–Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an 'integral' of the Schouten–Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.

The Drinfeld-Sokolov formalism is extended to the case of operator-valued affine Lie algebra to derive nonlinear integrable dynamical systems in (2+1) dimensions. The Poisson structure of these integrable equations are also worked out. While from the first- and second-order flows we get some new integrable equations in (2+1) dimensions, the KP equation is seen to result from the third-order flow. Complete integrability of such equations and the existence of the bi-Hamiltonian structure are demonstrated.

Some important properties of Poisson groupoids are discussed. In particular, we obtain a useful formula for the Poisson tensor of an arbitrary Poisson groupoid, which generalizes the well-known multiplicativity condition for Poisson groups. Morphisms between Poisson groupoids and between Lie bialgebroids are also discussed. In particular, for a special class of Lie bialgebroid morphisms, we give an explicit lifting construction. As an application, we prove that a Poisson group action on a Poisson manifold lifts to a Poisson action on its α-simply connected symplectic groupoid.

First we show that, associated to any Poisson vector field [Formula: see text] on a Poisson manifold [Formula: see text], there is a canonical Lie algebroid structure on the first jet bundle [Formula: see text] which, depends only on the cohomology class of [Formula: see text]. We then introduce the notion of a cosymplectic groupoid and we discuss the integrability of the first jet bundle into a cosymplectic groupoid. Finally, we give applications to Atiyah classes and [Formula: see text]-algebras.

A class of two dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang-Mills systems, is presented. Locally, the solutions of the classical equations of motion are given. A general scheme for the quantization of the models in a Hamiltonian formulation is found., Comment: 6 pages, LaTeX, TUW9403

It is shown that non-Abelian Thirring models can be formulated as the Hamiltonian systems with Poisson brackets of the Lie algebraic structure. This fact allows Thirring models to be quantized by the Hamiltonian method. We show that the classical Lie-Poisson structure can be promoted to the quantum level in two different ways corresponding to different phases of non-Abelian Thirring models. There are special values of coupling constants at which the Hamiltonian quantization of Thirring models can be carried out consistently with the conformal invariance. These fixed couplings appear to be the solutions of the Virasoro master equation.

The problem of defining Poisson brackets for a degenerate Lagrangian without introduction of canonical variables is discussed. More generally, we introduce and give a complete geometrical description of a class of Poisson brackets on a presymplectic manifold. The construction is illustrated by both finite-dimensional and field-theoretic examples.

The Poisson structure of Lax operator of B and C type and super Lax operator which has odd parity are studied. The algebraic structure of Poisson structure as a background of Lax equation is thrown light on.

The classical [Formula: see text] algebra Polyakov is shown to be equivalent to the second Poisson structure of a new integrable hierarchy of nonlinear equations. The hierarchy is related to the Boussinesq hierarchy by interhcanging the roles of the space and time variables x and t in the Boussinesq equation. From this relation the Miura map, relating the new hierarchy to its modified version, can be derived systematically. It is found to be equivalent to the known free field representation of the [Formula: see text] algebra.

Neumann’s model, describing the motion of a particle on an N-sphere under harmonic forces, is studied from the point of view of classical and quantum integrability. Classical integrability is derived from a generalized structure, “R-S couple” or “D-matrix” for the Poisson brackets of the Lax operator. The already-known set of conserved quantities for this model turns out to follow straightforwardly from this structure. It gives rise to a set of commuting operators at the quantum level, and the algebra of Lax operators directly follows from the classical one.

It is known that the topological T-duality exchanges $H$ and $F$-fluxes. In this paper, we reformulate the topological T-duality as an exchange of two Lie algebroids in the generalized tangent bundle. Then, we apply the same formulation to the Poisson-generalized geometry, which is introduced in arXiv:1408.2649 to define $R$-fluxes as field strength associated with $\beta$-transformations. We propose a definition of $Q$-flux associated with $\beta$-diffeomorphisms, and show that the topological T-duality exchanges $R$ and $Q$-fluxes., Comment: 30 pages

We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of $\beta$-diffeomorphisms and $\beta$-transformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac structures and generalized Riemannian structures. In particular, $R$-fluxes are formulated as a twisting of this Courant algebroid by a local $\beta$-transformations, in the same way as $H$-fluxes are the twist of the generalized tangent bundle. It is a $3$-vector classified by Poisson $3$-cohomology and it appears in a twisted bracket and in an exact sequence., Comment: 22 pages

The algebraic-geometrical solutions of three (2 + 1)-dimensional equations (including mKP equation and coupled mKP equation) are discussed by Hamiltonian approach. First, the Poisson structure on CN × RN is introduced to give a Hamiltonian system associated with the derivative nonlinear Schrödinger (DNLS) hierarchy. The Hamiltonian system is proved to be Liouville integrable, accordingly the solutions of three (2 + 1)-dimensional nonlinear equations can be solved by three compatible Hamiltonian flows. Second, the canonical separated variables and Hamilton–Jacobi theory is used to definite action-angle variables for Hamiltonian flows. At last, by Riemann–Jacobi inversion, the algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations are obtained. Besides, the algebraic-geometrical solutions of the first two DNLS equations are also given.

We describe the procedure for obtaining Hamiltonian equations on a manifold with $so(k, m)$ Lie-Poisson bracket from a variational problem. This implies identification of the manifold with base of a properly constructed fiber bundle embedded as a surface into the phase space with canonical Poisson bracket. Our geometric construction underlies the formalism used for construction of spinning particles in [24-27], and gives precise mathematical formulation of the oldest idea about spin as the "inner angular momentum"., published version

We discuss linear in momenta Poisson structure for the generalized nonholonomic Chaplygin sphere problem and prove that it is nontrivial deformation of the canonical Poisson structure on e*(3).

It is shown that any nonsingular Lagrangian describing the motion of a particle on a semisimple Lie group possesses a Fundamental Poisson bracket Relation (FPR) and consequently charges in involution. This property is independent of the dynamics of the model and can be derived in a quite simple and general way from the geometric and algebraic structures of the group manifold. The conditions a Hamiltonian has to satisfy in order those charges are to be conserved are discussed. These conditions lead to an algebra which plays an important role in the construction of conserved charges. In the second paper of the series, this work is extended to the coset spaces which are symmetric spaces.