Noncommutative Poisson Algebra Research Articles

Noncommutative branch‐cut quantum gravity with a self‐coupling inflation scalar field: Dynamical equations

AbstractThis article focuses on a recently developed formulation based on the noncommutative branch‐cut cosmology, the Wheeler‐DeWitt (WdW) equation, the Hořava–Lifshitz quantum gravity, chaotic and the coupling of the corresponding Lagrangian approach with the inflaton scalar field. Assuming a mini‐superspace of variables obeying the noncommutative Poisson algebra, we examine the impact of the inflaton scalar field on the evolutionary dynamics of the branch‐cut Universe scale factor, characterized by the dimensionless helix‐like function . This scale factor characterizes a Riemannian foliated spacetime that topologically overcomes the primordial singularities. We take the Hořava–Lifshitz action modeled by branch‐cut quantum gravity as our starting point, which depends on the scalar curvature of the branched Universe and its derivatives and which preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner formalism. We then investigate the sensitivity of the scale factor of the branch‐cut Universe's dynamics.

Noncommutative branch‐cut quantum gravity with a self‐coupling inflaton scalar field: The wave function of the Universe

AbstractThis article focuses on the implications of a noncommutative formulation of branch‐cut quantum gravity. Based on a mini‐superspace structure that obeys the noncommutative Poisson algebra, combined with the Wheeler–DeWitt equation and Hořava–Lifshitz quantum gravity, we explore the impact of a scalar field of the inflaton‐type in the evolution of the Universe's wave function. Taking as a starting point the Hořava–Lifshitz action, which depends on the scalar curvature of the branched Universe and its derivatives, the corresponding wave equations are derived and solved. The noncommutative quantum gravity approach adopted preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner Formalism. In this work we delve deeper into a mini‐superspace of noncommutative variables, incorporating scalar inflaton fields and exploring inflationary models, particularly chaotic and nonchaotic scenarios. We obtained solutions to the wave equations without resorting to numerical approximations. The results indicate that the noncommutative algebraic space captures low and high spacetime scales, driving the exponential acceleration of the Universe.

RETRACTED ARTICLE: Split Malcev-Poisson-Jordan algebras

We introduce the class of split Malcev-Poisson-Jordan algebras as the natural extension of the one of split Malcev Poisson algebras, and therefore split (non-commutative) Poisson algebras. We show that a split Malcev-Poisson-Jordan algebra P can be written as a direct sum with any Ij a non-zero ideal of P in such a way that satisfies for . Under certain conditions, it is shown that the above decomposition of P is by means of the family of its simple ideals.

Noncommutative Poisson bialgebras

In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O-operators on noncommutative Poisson algebras, and noncommutative pre-Poisson algebras are introduced, by which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some special noncommutative Poisson algebras obtained from these structures.

Noncommutative and Dynamical Analysis in a Curved Phase-space

In this work, we have analyzed the dynamics of the model of a free particle over a 2-sphere in a noncommutative (NC) phase-space. Besides, we have shown that the solution of the equations of motion allows one to show the equivalence between the movement of the particle upon a 2-sphere and the one described by a central field. We have considered the eective force felt by the particle as being caused by the curvature of the space. We have analyzed the NC Poisson algebra of classical observables in order to obtain the NC corrections to Newton’s second law analogous to the one caused by a central field. We have also discussed the relation between ane connection and Dirac brackets, as they describe the proper evolution of the model over the surface of constraints in the Lagrangian and Hamiltonian formalisms, respectively. As an application, we have treated the so-called Zitterbewegung of the Dirac electron. Since it is assumed to be an observable eect, then we have traced its physical origin by assuming that the electron has an internal structure.

The global extension problem, crossed products and co-flag non-commutative Poisson algebras

Let P be a Poisson algebra, E a vector space and π:E→P an epimorphism of vector spaces with V=Ker(π). The global extension problem asks for the classification of all Poisson algebra structures that can be defined on E such that π:E→P becomes a morphism of Poisson algebras. From a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component. All such Poisson algebra structures on E are classified by an explicitly constructed classifying set GPH2(P,V) which is the coproduct of all non-abelian cohomological objects PH2(P,(V,⋅V,[−,−]V)) which are the classifying sets for all extensions of P by (V,⋅V,[−,−]V). The second classical Poisson cohomology group H2(P,V) appears as the most elementary piece among all components of GPH2(P,V). Several examples are provided in the case of metabelian Poisson algebras or co-flag Poisson algebras over P: the latter being Poisson algebras Q which admit a finite chain of epimorphisms of Poisson algebras Pn:=Q⟶πnPn−1⋯P1⟶π1P0:=P such that dim⁡(Ker(πi))=1, for all i=1,⋯,n.

Left-right noncommutative Poisson algebras

AbstractThe notions of left-right noncommutative Poisson algebra (NPlr-algebra) and left-right algebra with bracket AWBlr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NPlr-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWBlr and NPlr-algebras the notions of actions, representations, centers, actors and crossed modules are described as special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NPlr-algebras and AWBlr (resp. of NPr-algebras and AWBr) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWBr, the Hochschild or/and Leibniz cohomological dimension of P is ≤ n are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.

Non-Commutative Poisson Algebra Structures on Lie Algebra [formula omitted] with Nullity M

Non-commutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln(ℂ˜q) are determined.

Noncommutative Poisson brackets on Loday algebras and related deformation quantization

Given a Lie algebra, there uniquely exists a Poisson algebra which is called a Lie-Poisson algebra over the Lie algebra. We will prove that given a Loday/Leibniz algebra there exists uniquely a noncommutative Poisson algebra over the Loday algebra. The noncommutative Poisson algebras are called the Loday-Poisson algebras. In the super/graded cases, the Loday-Poisson bracket is regarded as a noncommutative version of classical (linear) Schouten-Nijenhuis bracket. It will be shown that the Loday-Poisson algebras form a special subclass of Aguiar's dual-prePoisson algebras. We also study a problem of deformation quantization over the Loday-Poisson algebra. It will be shown that the polynomial Loday-Poisson algebra is deformation quantizable and that the associated quantum algebra is Loday's associative dialgebra.

(Quasi-)Poisson enveloping algebras

We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.

On the structure of split non-commutative Poisson algebras

A non-commutative Poisson algebra is a Lie algebra endowed with an, not necessarily commutative, associative product in such a way that the Lie and associative products are compatible via the Leibniz identity. If we part from a split Lie algebra, with respect to a maximal abelian subalgebra H, we obtain a so-called split non-commutative Poisson algebra. This article is devoted to the study of the structure of split non-commutative Poisson algebras, by showing that any of such an algebra 𝒫 is of the form with U a linear subspace of H and any a well-described ideal of 𝒫, satisfying if j ≠ k. Under certain conditions the simplicity of 𝒫 is characterized and it is shown that 𝒫 is the direct sum of the family of its simple ideals.

Noncommutative Hamiltonian dynamics on foliated manifolds

First, we review the notion of a Poisson structure on a noncommutative algebra due to Block, Getzler and Xu, and we introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and construct a class of Hamiltonian vector fields associated with this Poisson structure.

Non-commutative Poisson Algebra Structures on the Lie Algebra $so_n\widetilde{({\Bbb C}_Q)}$

Non-commutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on [Formula: see text] are determined.

Poisson algebra structures on sp 2l( ) with nullity m

Noncommutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this article, the noncommutative Poisson algebra structures on sp 2l( ▪) are determined.

Transition from AdS universe to DS universe in the BPP model

It can be shown that in the BPP model the smooth phase transition from the asymptotically decelerated AdS universe to the asymptotically accelerated DS universe is possible by solving the modified semiclassical equations of motion. This transition comes from noncommutative Poisson algebra, which gives the constant curvature scalars asymptotically. The decelerated expansion of the early universe is due to the negative energy density with the negative pressure induced by quantum back reaction, and the accelerated late-time universe comes from the positive energy and the negative pressure which behave like dark energy source in recent cosmological models.

Noncommutative geometry of Poisson structures

Geometric, algebraic, and homological properties of Poisson structures on smooth manifolds are studied. Noncommutative (NC) foundations of these structures are introduced for associative Poisson algebras; noncommutative generalizations of such notions of the classical symplectic geometry as a degenerate Poisson structure, a Poisson submanifold, a symplectic foliation, and a symplectic leaf for associative Poisson algebras are also introduced. These structures are considered for the case of the endomorphism algebra of a vector bundle, and a full description of the family of Poisson structures for this algebra is given. An algebraic construction of the reduction procedure for degenerate noncommutative Poisson structures is developed. An NC generalization of the Bott connection on a foliated manifold is introduced by using the notions of NC submanifold and quotient manifold. This definition is applied to degenerate NC Poisson algebras, which allows one to consider the Bott connection not only for regular but also for singular Poisson structures.

Future-singularity-free accelerating expansion with modified Poisson brackets

We show that the second accelerating expansion of the universe appears smoothly from the decelerating phase, which follows the initial inflation, in the two-dimensional soluble semiclassical dilaton gravity along with the modified Poisson brackets with noncommutativity between the relevant fields. This is in contrast to the fact that the ordinary solution of the equations of motion following from the conventional Poisson algebra describes a permanent accelerating universe without any phase change. In this modified model, it turns out that the noncommutative Poisson algebra is responsible for the remarkable phase transition to the second accelerating expansion.

DEFORMATION QUANTIZATION OF PSEUDO-SYMPLECTIC (POISSON) GROUPOIDS

We introduce a new kind of groupoid—a pseudo-etale groupoid, which provides many interesting examples of noncommutative Poisson algebras as defined by Block, Getzler, and Xu. Following the idea that symplectic and Poisson geometries are the semiclassical limits of the corresponding quantum geometries, we quantize these noncommutative Poisson algebras in the framework of deformation quantization.

Noncommutative Classical Mechanics

In this work, I investigate the noncommutative Poisson algebra of classical observables corresponding to a proposed general Noncommutative Quantum Mechanics, \cite{1}. I treat some classical systems with various potentials and some Physical interpretations are given concerning the presence of noncommutativity at large scales (Celeste Mechanics) directly tied to the one present at small scales (Quantum Mechanics) and its possible relation with UV/IR mixing.

FINITE-DIMENSIONAL NON-COMMUTATIVE POISSON ALGEBRAS. II

Here is a structure theorem of a finite-dimensional non-commutative Poisson algebra A. A nice element ε of A will be found, so that the Lie module action of an element of a large Poisson subalgebra of A on A is described in terms of ε and the ordinary associative commutator. Consequently, we can figure out a structure of A when the Jacobson radical rad A satisfies (rad A)2 = 0. This structure theorem leads us to a classification of the finite-dimensional simple Poisson A-modules.