Quadratic Poisson Structures Research Articles

Twists of graded Poisson algebras and related properties

We introduce a Poisson version of the graded twist of a graded associative algebra and prove that every graded Poisson structure on a connected graded polynomial ring A:=k[x1,…,xn] is a graded twist of a unimodular Poisson structure on A, namely, if π is a graded Poisson structure on A, then π has a decompositionπ=πunim+1∑i=1ndeg⁡xiE∧m where E is the Euler derivation, πunim is the unimodular graded Poisson structure on A corresponding to π, and m is the modular derivation of (A,π). This result is a generalization of the same one in the quadratic setting. The rigidity of graded twisting, PH1-minimality, and H-ozoneness are studied. As an application, we compute the Poisson cohomologies of the quadratic Poisson structures on the polynomial ring of three variables when the potential is irreducible, but not necessarily having an isolated singularity.

A note on quadratic Poisson brackets on mathrm {gl}(n,{mathbb {R}}) related to Toda lattices

It is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic r-matrix Poisson brackets on the associative algebra mathrm {gl}(n,{mathbb {R}}). We here show that the quadratic bracket on mathrm {gl}(n,{mathbb {R}}), corresponding to the r-matrix defined by the splitting of mathrm {gl}(n,{mathbb {R}}) into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle T^* mathrm {GL}(n,{mathbb {R}}). This complements the interpretation of the linear r-matrix bracket as a reduction of the canonical Poisson bracket of the cotangent bundle.

Kahan Discretizations of Skew-Symmetric Lotka-Volterra Systems and Poisson Maps

The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph Γ, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic Poisson structure of the Lotka-Volterra system, we say that the graph Γ has the Kahan-Poisson property. We show that if Γ is connected, it has the Kahan-Poisson property if and only if it is a cloning of a graph with vertices $1,2,\dots ,n$ , with an arc i → j precisely when i < j, and with all arcs having the same value. We also prove a similar result for augmented graphs, which correspond with deformed Lotka-Volterra systems and show that the obtained Lotka-Volterra systems and their Kahan discretizations are superintegrable as well as Liouville integrable.

The quadratic Poisson structures and related nonassociative noncommutative Zinbiel type algebras

There are studied algebraic properties of the quadratic Poisson brackets on nonassociative noncommutive algebras, compatible with their multiplicative structure. Their relations both with differentiations of the symmetric tensor algebras and Yang-Baxter structures on the adjacent Lie algebras are demonstrated. Special attention is payed to the quadtatic Poisson brackets of the Lie-Poisson type, the examples of the Novikov and Leibniz algebras are discused. The nonassociated structures of commutative algebras related with Novikov, Leibniz, Lie and Zinbiel algebras are studied in details

Poisson structures for difference equations

We study the existence of log-canonical Poisson structures that are preserved by difference equations of a special form. We also study the inverse problem, given a log-canonical Poisson structure to find a difference equation preserving this structure. We give examples of quadratic Poisson structures that arise for the Kadomtsev–Petviashvili (KP) type maps which follow from a travelling-wave reduction of the corresponding integrable partial difference equation.

Rigidity of Quadratic Poisson Tori

We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of ${\mathbb{N}}$-graded connected cluster algebras equipped with Gekhtman-Shapiro-Vainshtein Poisson structures. Based on this, we classify the groups of algebraic Poisson automorphisms of the open Schubert cells of the full flag varieties of semisimple algebraic groups over fields of characteristic 0, equipped with the standard Poisson structures. Their coordinate rings can be identified with the semiclassical limits of the positive parts $U_q({\mathfrak{n}}_+)$ of the quantized universal enveloping algebras of semisimple Lie algebras, and the last result establishes a Poisson version of the Andruskiewitsch-Dumas conjecture on ${\mathrm{Aut}} U_q({\mathfrak{n}}_+)$.

On a Poisson homogeneous space of bilinear forms with a Poisson–Lie action

Let be the space of bilinear forms on with defining matrices endowed with a quadratic Poisson structure of reflection equation type. The paper begins with a short description of previous studies of the structure, and then this structure is extended to systems of bilinear forms whose dynamics is governed by the natural action of the Poisson–Lie group on . A classification is given of all possible quadratic brackets on preserving the Poisson property of the action, thus endowing with the structure of a Poisson homogeneous space. Besides the product Poisson structure on , there are two other (mutually dual) structures, which (unlike the product Poisson structure) admit reductions by the Dirac procedure to a space of bilinear forms with block upper triangular defining matrices. Further generalisations of this construction are considered, to triples with the Poisson action , and it is shown that then acquires the structure of a Poisson symmetric space. Generalisations to chains of transformations and to the quantum and quantum affine algebras are investigated, as well as the relations between constructions of Poisson symmetric spaces and the Poisson groupoid. Bibliography: 30 titles.

Quantum deformations of projective three-space

We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi–Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has exactly six irreducible components, and we give explicit presentations for the generic members of each family in terms of generators and relations. The proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic Poisson structures in four dimensions, which we extract from Cerveau and Lins Neto's classification of degree-two foliations on projective space. Corresponding to the “exceptional” component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical Schwarzenberger bundles.

Twisted Poincaré duality between Poisson homology and Poisson cohomology

A version of the twisted Poincaré duality is proved between the Poisson homology and cohomology of a polynomial Poisson algebra with values in an arbitrary Poisson module. The duality is achieved by twisting the Poisson module structure in a canonical way, which is constructed from the modular derivation. In the case that the Poisson structure is unimodular, the twisted Poincaré duality reduces to the Poincaré duality in usual sense. The main result generalizes the work of Launois and Richard [8] for the quadratic Poisson structures and Zhu [25] for the linear Poisson structures.

Compatible quadratic Poisson brackets related to a family of elliptic curves

We construct nine pairwise compatible quadratic Poisson structures such that a generic linear combination of them is associated with an elliptic algebra in n generators. Explicit formulas for Casimir elements of this elliptic Poisson structure are obtained.

Quivers and Poisson structures

We produce natural quadratic Poisson structures on moduli spaces of representations of quivers. In particular, we study a natural Poisson structure for the generalised Kronecker quiver with 3 arrows.

Non-coboundary Poisson–Lie structures on the book group

All possible Poisson–Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Their classification is fully performed by relating these PL groups to the corresponding Lie bialgebra structures on the corresponding ‘book’ Lie algebra. By construction, all these Poisson structures are quadratic Poisson–Hopf algebras for which the group multiplication is a Poisson map. In contrast to the case of simple Lie groups, it turns out that most of the PL structures on the book group are non-coboundary ones. Moreover, from the viewpoint of Poisson dynamics, the most interesting PL book structures are just some of these non-coboundaries, which are explicitly analysed. In particular, we show that the two different q-deformed Poisson versions of the sl(2, R) algebra appear as two distinguished cases in this classification, as well as the quadratic Poisson structure that underlies the integrability of a large class of 3D Lotka–Volterra equations. Finally, the quantization problem for these PL groups is sketched.

Integrable deformations of Lotka–Volterra systems

The Hamiltonian structure of a class of three-dimensional (3D) Lotka–Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson–Lie group. As a consequence, the Poisson coalgebra map Δ(2) that is given by the group multiplication provides the keystone for the explicit construction of a new family of 3N-dimensional integrable systems that, under certain constraints, contain N sets of deformed versions of the 3D LV equations. Moreover, by considering the most generic Poisson–Lie structure on this group, a new two-parametric integrable perturbation of the 3D LV system through polynomial and rational perturbation terms is explicitly found.

Toric Poisson Structures

Let X(\Sigma) be a smooth projective toric variety for a complex torus T_\C. In this paper, a real T_\C-invariant Poisson structure \Pi_\Sigma is constructed on the complex manifold X(\Sigma), the symplectic leaves of which are the T_\C-orbits in X(\Sigma). It is shown that each leaf admits a Hamiltonian action by a sub-torus of the compact torus T\subset T_\C. However, the global action of T_\C on (X(\Sigma),\Pi_\Sigma) is Poisson but not Hamiltonian. The main result of the paper is a lower bound for the first Poisson cohomology of these structures. For the simplest case, X(\Sigma)=\CP^1, the Poisson cohomology is computed using a Mayer-Vietoris argument and known results on planar quadratic Poisson structures and in the example the bound is optimal. The paper concludes with the example of \CP^n, where the modular vector field with respect to a particular Delzant Liouville form admits a curious formula in terms of Delzant moment data. This formula enables one to compute the zero locus of this modular vector field and relate it to the Euclidean geometry of the moment simplex.

Bimodules and branes in deformation quantization

AbstractWe prove a version of Kontsevich’s formality theorem for two subspaces (branes) of a vector space X. The result implies, in particular, that the Kontsevich deformation quantizations of S(X*) and ∧(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet’s recent paper on Koszul duality in deformation quantization.

Completeness of Determinantal Hamiltonian Flows on the Matrix Affine Poisson Space

The matrix affine Poisson space (M_{m,n}, pi_{m,n}) is the space of complex rectangular matrices equipped with a canonical quadratic Poisson structure which in the square case m=n reduces to the standard Poisson structure on GL_n(C). We prove that the Hamiltonian flows of all minors are complete. As a corollary we obtain that all Kogan-Zelevinsky integrable systems on M_{n,n} are complete and thus induce (analytic) Hamiltonian actions of C^{n(n-1)/2} on (M_{n,n}, pi_{n,n}) (as well as on GL_n(C) and on SL_n(C)). We define Gelfand-Zeitlin integrable systems on (M_{n,n}, pi_{n,n}) from chains of Poisson projections and prove that their flows are also complete. This is an analog for the quadratic Poisson structure pi_{n,n} of the recent result of Kostant and Wallach [KW] that the flows of the complexified classical Gelfand-Zeitlin integrable systems are complete.

On bi-hamiltonian structure of some integrable systems on so* (4)

We classify quadratic Poisson structures on so*(4) and e*(3), which have the same foliations by symplectic leaves as canonical Lie-Poisson tensors. The separated variables for some of the corresponding bi-integrable systems are constructed.

Quadratic Poisson structures and Nambu mechanics

We study a special class of Poisson structures, namely quadratic Poisson structures, on R n which contains the celebrated Sklyanin algebras which play the significant role in the theory of exactly solvable models of classical and quantum systems. In this paper we study such quadratic Poisson structures via Nambu mechanics.

On a general approach to the formal cohomology of quadratic Poisson structures

We propose a general approach to the formal Poisson cohomology of r -matrix induced quadratic structures, we apply this device to compute the cohomology of structure 2 of the Dufour–Haraki classification, and provide complete results also for the cohomology of structure 7.

Internal deformation of Lie algebroids and symplectic realizations

Given a Lie algebroid and a bundle over its base which is endowed with a localizable Poisson structure and a flat connection, we construct an extended bundle whose dual is endowed with an almost-Poisson structure that is a quadratic Poisson structure when a certain compatibility property is satisfied. This new formalism on Lie algebroids describes systems with internal degrees of freedom.