相关论文: Integer solutions of integral inequalities and H-i…
We study different algebraic and geometric properties of Heisenberg invariant Poisson polynomial quadratic algebras. We show that these algebras are unimodular. The elliptic Sklyanin-Odesskii-Feigin Poisson algebras $q_{n,k}(\mathcal E)$…
Some aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of view, due to the…
A new family of solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is investigated. This family is mathematically remarkable, as the functional dependences of the solutions appear to be associated…
A systematic investigation of the skew-symmetric solutions of the three-dimensional Jacobi equations is presented. As a result, three disjoint and complementary new families of solutions are characterized. Such families are very general,…
An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and non-local families of R-matrix solutions to the modified Yang-Baxter equation. The…
An n-dimensional solution family of the Jacobi equations is characterized and investigated, including the global determination of its main features: the Casimir invariants, the construction of the Darboux canonical form and the proof of…
The unimodularity condition for a Poisson structure (ie., a Poisson structure with a trivial modular class) induces a Poincar\'e duality between its Poisson homology and its Poisson cohomology. Therefore an information about the Poisson…
A new four-dimensional family of skew-symmetric solutions of the Jacobi equations for Poisson structures is characterized. As a consequence, previously known types of Poisson structures found in a diversity of physical situations appear to…
A new family of skew-symmetric solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is characterized and analyzed. Such family has some remarkable properties. In first place, it is defined for…
We address the problem of classifying discrete differential-geometric Poisson brackets (dDGPBs) of any fixed order on target space of dimension 1. It is proved that these Poisson brackets (PBs) are in one-to-one correspondence with the…
We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an…
In this paper, we study invariant Poisson structures on homogeneous manifolds, which serve as a natural generalization of homogeneous symplectic manifolds previously explored in the literature. Our work begins by providing an algebraic…
Two Poisson structures invariant with respect to discrete transformation of the Maximal root in the case of arbitrary semi-simple algebras are presented in explicit form. Thus the problem of construction of equations of n-wave hierarchy in…
In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian…
We show that the space of observables of test particles carries a natural Jacobi structure which is manifestly invariant under the action of the Poincar\'{e} group. Poisson algebras may be obtained by imposing further requirements. A…
Let $G\subset SO(4)$ denote a finite subgroup containing the Heisenberg group. In these notes we classify all these groups, we find the dimension of the spaces of $G$-invariant polynomials and we give equations for the generators whenever…
A family of solutions of the Jacobi PDEs is investigated. This family is $n$-dimensional, of arbitrary nonlinearity and can be globally analyzed (thus improving the usual local scope of Darboux theorem). As an outcome of this analysis it is…
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…
Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation…
In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson algebras and vertex algebras) and…