Poisson Algebra Research Articles

Fundamental Theorem of Poisson (A,H)-Hopf modules

Let k be a field, H a Hopf algebra with a bijective antipode and A an H-comodule Poisson algebra. We assume that there is an H-colinear algebra map from H to the Poisson-center of A. We generalize the Fundamental Theorem of (A,H)-Hopf modules to Poisson (A,H)-Hopf modules, and we deduce relative projectivity in the category of Poisson (A,H)-Hopf modules. In many applications, A could be a rational G-module Poisson algebra, where G is an affine algebraic group or the coordinate ring of an affine Poisson variety over C on which acts morphically an algebraic group via automorphisms of Poisson varieties.

Topology of the isoenergy manifolds of the Kirchhoff rigid body case on e(3)

Our aim is to study topology of the Kirchhoff case of rigid body motion in an ideal incompressible fluid. We introduce the reduced potential for general Hamiltonian systems on e(3) with mixed quadratic terms. In application to the Kirchhoff case, we describe the Reeb graphs of the reduced potential. We provide a complete topological description of the three-dimensional isoenergy manifolds for that system, based on a combinatorial study of the Reeb graphs. Studying its momentum map, we describe the points of ranks zero and one. The Poincaré model of a rigid body with an ellipsoidal cavity filled with an ideal incompressible liquid has a Hamiltonian of the same form as the Kirchhoff Hamiltonian, with the underlying Poisson algebra being so(4). A similar analysis of bifurcations of the momentum map is presented in the Poincaré case as well.

Poisson algebra structure on the invariants of pairs of matrices of degree three

We provide a table of multiplication of the Poisson algebra on the minimal set of generators of the invariants of pairs of matrices of degree three.

A gravitational action with stringy Q and R fluxes via deformed differential graded Poisson algebras

We study a deformation of a 2-graded Poisson algebra where the functions of the phase space variables are complemented by linear functions of parity odd velocities. The deformation is carried by a 2-form B-field and a bivector Π, that we consider as gauge fields of the geometric and non-geometric fluxes H, f, Q and R arising in the context of string theory compactification. The technique used to deform the Poisson brackets is widely known for the point particle interacting with a U(1) gauge field, but not in the case of non-abelian or higher spin fields. The construction is closely related to Generalized Geometry: with an element of the algebra that squares to zero, the graded symplectic picture is equivalent to an exact Courant algebroid over the generalized tangent bundle E ≅ TM ⊕ T∗M, and to its higher gauge theory. A particular idempotent graded canonical transformation is equivalent to the generalized metric. Focusing on the generalized differential geometry side we construct an action functional with the Ricci tensor of a connection on covectors, encoding the dynamics of a gravitational theory for a contravariant metric tensor and Q and R fluxes. We also extract a connection on vector fields and determine a non-symmetric metric gravity theory involving a metric and H-flux.

-(bi)derivations and transposed Poisson algebra structures on Lie algebras

In the present paper, we introduce the notion of a δ-biderivation. First, we provide some properties of δ-biderivations and illustrate their applications. In particular, we establish a close relationship between -biderivations and transposed Poisson algebras. Second, we compute -derivations on the twisted Heisenberg–Virasoro, Schrödinger–Virasoro, extended Schrödinger–Virasoro and twisted Schrödinger–Virasoro algebras, respectively. It turns out that they have no nontrivial -derivations. Hence they have neither nonzero -biderivations nor nontrivial transposed Poisson algebra structures. Third, we classify transposed Poisson algebra structures on the Heisenberg and some current Lie algebras. This enables us to provide examples of Lie algebras having nontrivial transposed Poisson algebra structures.

Symmetric polynomials in the free metabelian Poisson algebras

Let K be a field of characteristic zero and [Formula: see text] be a finite set of variables. Consider the free metabelian Poisson algebra [Formula: see text] of rank n generated by [Formula: see text] over K. An element in [Formula: see text] is called symmetric if it is preserved under any change of variables, i.e. under the action of each permutation in [Formula: see text]. In this study, we determine the algebra [Formula: see text] of symmetric polynomials of [Formula: see text].

The replicator dynamics of zero-sum games arise from a novel poisson algebra

We show that the replicator dynamics for zero-sum games arises as a result of a non-canonical bracket that is a hybrid between a Poisson Bracket and a Nambu Bracket. The resulting non-canonical bracket is parameterized both the by the skew-symmetric payoff matrix and a mediating function. The mediating function is only sometimes a conserved quantity, but plays a critical role in the determination of the dynamics. As a by-product, we show that for the replicator dynamics this function arises in the definition of a natural metric on which phase flow-volume is preserved. Additionally, we show that the non-canonical bracket satisfies all the same identities as the Poisson bracket except for the Jacobi identity (JI), which is satisfied for special cases of the mediating function. In particular, the mediating function that gives rise to the replicator dynamics yields a bracket that satisfies JI. This neatly explains why the mediating function allows us to derive a metric on which phase flow is conserved and suggests a natural geometry for zero-sum games that extends the Symplectic geometry of the Poisson bracket and potentially an alternate approach to quantizing evolutionary games.

Graded Poisson algebras on bordism groups of garlands

Let Mm be an oriented manifold and let N be a set consisting of oriented closed manifolds of the same odd dimension n. We consider a topological space GN,M of commutative diagrams. Each commutative diagram consists of a few manifolds from N that are mapped to M and of one point spaces pt that are each mapped to a pair of manifolds from N.We consider the oriented bordism group Ω⁎(GN,M)=⊕i=0∞Ωi(GN,M). We introduce the operations ⋆ and [⋅,⋅] on Ω⁎(GN,M)⊗Q, that make Ω⁎(GN,M)⊗Q into a Z-graded (m−2n)-Poisson algebra.For N={S1} and a surface M=F2, the subalgebra Ω0(G{S1},F2)⊗Q of our algebra is related to the Andersen-Mattes-Reshetikhin Poisson algebra of chord-diagrams.

Admissible Poisson bialgebras

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommutative and one anti-cocommutative comultiplications generalizes and illustrates the polarization–depolarization process in the context of bialgebras. The study of a special class of adm-Poisson bialgebras which include the known coboundary Poisson bialgebras in [27] as a proper subclass in general, illustrating an advantage in terms of the presentation with one operation, leads to the introduction of adm-Poisson Yang–Baxter equation in an adm-Poisson algebra. It is an unexpected consequence that both the adm-Poisson Yang–Baxter equation and the associative Yang–Baxter equation have the same form and thus it motivates and simplifies the involved study from the study of the associative Yang–Baxter equation, which is another advantage in terms of the presentation with one operation. A skew-symmetric solution of adm-Poisson Yang–Baxter equation gives an adm-Poisson bialgebra. Finally, the notions of an [Formula: see text]-operator of an adm-Poisson algebra and a pre-adm-Poisson algebra are introduced to construct skew-symmetric solutions of adm-Poisson Yang–Baxter equation and hence adm-Poisson bialgebras. Note that a pre-adm-Poisson algebra gives an equivalent presentation for a pre-Poisson algebra introduced by Aguiar.

Brane current algebras and generalised geometry from QP manifolds. Or, \u201cwhen they go high, we go low\u201d

We construct a Poisson algebra of brane currents from a QP-manifold, and show their Poisson brackets take a universal geometric form. This generalises a result of Alekseev and Strobl on string currents and generalised geometry to include branes with worldvolume gauge fields, such as the D3 and M5. Our result yields a universal expression for the ’t Hooft anomaly that afflicts isometries in the presence of fluxes. We determine the current algebra in terms of (exceptional) generalised geometry, and show that the tensor hierarchy gives rise to a brane current hierarchy. Exceptional complex structures produce pairs of anomaly-free current subalgebras on the M5-brane worldvolume.

Reductions of Invariant bi-Poisson Structures and Locally Free Actions

Let (X,G,ω1,ω2,{ηt}) be a manifold with a bi-Poisson structure {ηt} generated by a pair of G-invariant symplectic structures ω1 and ω2, where a Lie group G acts properly on X. We prove that there exists two canonically defined manifolds (RLi,Gi,ω1i,ω2i,{ηit}), i=1,2 such that (1) RLi is a submanifold of an open dense subset X(H)⊂X; (2) symplectic structures ω1i and ω2i, generating a bi-Poisson structure {ηit}, are Gi- invariant and coincide with restrictions ω1|RLi and ω2|RLi; (3) the canonically defined group Gi acts properly and locally freely on RLi; (4) orbit spaces X(H)/G and RLi/Gi are canonically diffeomorphic smooth manifolds; (5) spaces of G-invariant functions on X(H) and Gi-invariant functions on RLi are isomorphic as Poisson algebras with the bi-Poisson structures {ηt} and {ηit} respectively. The second Poisson algebra of functions can be treated as the reduction of the first one with respect to a locally free action of a symmetry group.

The infinite dimensional unital Nambu–Poisson algebra of order 3

From a commutative associative algebra with a basis the infinite dimensional unital Nambu–Poisson algebra of order 3 is constructed, which is also a canonical Nambu–Poisson algebra, and its structures and derivations are discussed. It is proved that: (1) there is a minimal set of generators of consisting of six vectors; (2) the quotient 3-Lie algebra is simple; (3) four infinite dimensional 3-Lie algebras: the 3-Virasoro–Witt algebra (), and the 3-algebra can be embedded in the unital Nambu–Poisson algebra of order 3.

Poisson cohomology, Koszul duality, and Batalin–Vilkovisky algebras

We study the noncommutative Poincaré duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich’s deformation quantization as well as Koszul duality preserve the corresponding Poincaré duality. As a corollary, the Batalin– Vilkovisky algebra structures that naturally arise in these cases are all isomorphic.

Reflection Groups and Rigidity of Quadratic Poisson Algebras

In this paper, we study the invariant theory of quadratic Poisson algebras. Let G be a finite group of the graded Poisson automorphisms of a quadratic Poisson algebra A. When the Poisson bracket of A is skew-symmetric, a Poisson version of the Shephard-Todd-Chevalley theorem is proved stating that the fixed Poisson subring AG is skew-symmetric if and only if G is generated by reflections. For many other well-known families of quadratic Poisson algebras, we show that G contains limited or even no reflections. This kind of Poisson rigidity result ensures that the corresponding fixed Poisson subring AG is not isomorphic to A as Poisson algebras unless G is trivial.

Double Quasi-Poisson Algebras are Pre-Calabi-Yau

Abstract In this article, we prove that double quasi-Poisson algebras, which are noncommutative analogues of quasi-Poisson manifolds, naturally give rise to pre-Calabi-Yau algebras. This extends one of the main results in [11], where a correspondence between certain pre-Calabi-Yau algebras and double Poisson algebras was found (see also [13, 12, 10]). However, a major difference between the pre-Calabi-Yau algebra constructed in the mentioned articles and the one constructed in this work is that the higher multiplications indexed by even integers of the underlying $A_{\infty }$-algebra structure of the pre-Calabi-Yau algebra associated with double quasi-Poisson algebra do not vanish, but are given by nice cyclic expressions multiplied by explicitly determined coefficients involving the Bernoulli numbers.

Noncommutative elliptic Poisson structures on projective spaces

We review noncommutative Poisson structures on affine and projective spaces over C. We also construct a class of examples of noncommutative Poisson structures on CPn−1 for n>2. These noncommutative Poisson structures depend on a modular parameter τ∈C and an additional discrete parameter k∈Z, where 1≤k<n and k,n are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras qn,k(τ).

On analogs of some classical group-theoretic results in Poisson algebras

We investigate the Poisson algebras, in which the n-th hypercenter (center) has a finite codimension. It was established that, in this case, the Poisson algebra P includes a finite-dimensional ideal K such that P/K is nilpotent (Abelian). Moreover, if the n-th hypercenter of a Poisson algebra P over some field has a finite codimension, and if P does not contain zero divisors, then P is Abelian.

Poisson primitive ideals of a Poisson order

Let R be an affine Poisson algebra over a field of characteristic zero and let A be a Poisson order over R. It is proved that every Poisson primitive ideal of A is annihilator of a simple Poisson A-module.

$$\frac{1}{2}$$-derivations of Lie algebras and transposed Poisson algebras

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras was established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, algebra $\mathcal{W}(a,-1)$, thin Lie algebra and solvable Lie algebra with abelian nilpotent radical) were constructed. In particular, we constructed an example of the transposed Poisson algebra with associative and Lie parts isomorphic to the Laurent polynomials and the Witt algebra. On the other side, it was proven that there are no non-trivial transposed Poisson algebras with Lie algebra part isomorphic to a semisimple finite-dimensional algebra, a simple finite-dimensional superalgebra, the Virasoro algebra, $N=1$ and $N=2$ superconformal algebras, or a semisimple finite-dimensional $n$-Lie algebra.

Duals of Semisimple Poisson–Lie Groups and Cluster Theory of Moduli Spaces of G-local Systems

Abstract We study the dual $\textrm{G}^{\ast }$ of a standard semisimple Poisson–Lie group $\textrm{G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}(\textrm{G}^{\ast })$ can be naturally embedded into a quotient algebra of a cluster Poisson algebra with a Weyl group action. The coordinate ring $\mathcal{O}(\textrm{G}^{\ast })$ admits a natural basis, which has positive integer structure coefficients and satisfies an invariance property under a braid group action. We continue the study of the moduli space $\mathscr{P}_{\textrm{G},{{\mathbb{S}}}}$ of $\textrm{G}$-local systems introduced in [ 16] and prove that the coordinate ring of $\mathscr{P}_{\textrm{G}, {{\mathbb{S}}}}$ coincides with its underlying cluster Poisson algebra.