Noncommutative Extension Research Articles

The quasi-Gramian solution of a non-commutative extension of the higher-order nonlinear Schrödinger equation

The nonlinear Schrödinger (NLS) equation, which incorporates higher-order dispersive terms, is widely employed in the theoretical analysis of various physical phenomena. In this study, we explore the non-commutative extension of the higher-order NLS equation. We treat real or complex-valued functions, such as g 1 = g 1(x, t) and g 2 = g 2(x, t) as non-commutative, and employ the Lax pair associated with the evolution equation, as in the commutation case. We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation. The soliton solutions are presented explicitly within the framework of quasideterminants. To visually understand the dynamics and solutions in the given example, we also provide simulations illustrating the associated profiles. Moreover, the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns within the context of modulational instability.

On The Group of Units Problem of the Non-Commutative Logical Extension of the Rings 𝒁𝒑 and 𝒁𝟐𝒏

This paper is dedicated to studying the group of units problem of the non-commutative logical extension of two different rings 𝑍𝑝 and 𝑍2𝑛, where we classify the group of units of these rings as semi-direct products of well-known abelian groups as follows: 𝑈(𝑁𝐶𝑅)𝑍𝑝≅𝑍𝑃−1∝(𝑍𝑃∝𝑍𝑃−1)𝑈(𝑁𝐶𝑅)2𝑛≅(𝑍2×𝑍2𝑛−2)⋉(𝑍2𝑛⋉(𝑍2×𝑍2𝑛−2)).

Is a loopy and noncommutative early Universe viable?

We conduct a (pre-)inflationary dynamics study of the chaotic quadratic potential within the framework of a simple noncommutative extension of effective loop quantum cosmology put forward recently. It is concluded that, for typically small values of the noncommutativity parameter, the resulting (pre-)inflationary scenario is essentially indistinguishable from the one corresponding to standard loop quantum cosmology.

On retracts determinating commutative trusses

This paper is devoted to study some relationships of trusses with rings. We characterize abelian groups A such that every truss with retract A has exactly two non-commutative multiplications. We describe commutative rings with nontrivial multiplication having non-commutative extension by the ring of integers.

A noncommutative extension of Mahler’s interpolation theorem

We prove a noncommutative generalization of Mahler’s theorem on interpolation series, a celebrated result of p -adic analysis. Mahler’s original result states that a function from \mathbb{N} to \mathbb{Z} is uniformly continuous for the p -adic metric d_p if and only if it can be uniformly approximated by polynomial functions. We prove an analogous result for functions from a free monoid A to a free group F(B) where d_p is replaced by the pro- p metric.

SELMER GROUPS OF ELLIPTIC CURVES OVER THE EXTENSION

AbstractIwasawa theory of elliptic curves over noncommutative $GL(2)$ extension has been a fruitful area of research. Over such a noncommutative p-adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the $PGL(2)$ extension and connect it with Iwasawa theory over the $GL(2)$ extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the $GL(2)$ extension being torsion is related with that of the $PGL(2)$ extension.

A Matrix Model of Four-Dimensional Noncommutative Gravity

In this review, we revisit our latest works regarding the description of the gravitational interaction on noncommutative spaces as matrix models. Specifically, inspired by the gauge-theoretic approach of (ordinary) gravity, we make use of the suggested methodology, modified appropriately for the noncommutative framework, of the well-established formulation of gauge theories on them. Making use of a covariant four-dimensional fuzzy space, we formulate the gauge theory with an extended gauge group due to noncommutativity. In turn, in order to decrease the amount of symmetry we employ a symmetry breaking and result with an action which describes a theory that is a minimal noncommutative extension of the original.

The dynamic of quantum entanglement of two dimensional harmonic oscillator in non-commutative space

In the present paper, we study the influence of non-commutativity on entanglement in a system of two oscillators-modes in interaction with its environment. The considered system is a two-dimensional harmonic oscillator in non-commuting spatial coordinates coupled to its environment. The dynamics of the covariance matrix, the separability criteria for two Gaussian states in non-commutative space coordinates, and the logarithmic negativity are used to evaluate the quantum entanglement in the system, which is compared to the commutative space coordinates case. The result is applied for two initially entangled states, namely the squeezed vacuum and squeezed thermal states. It can be observed that the phenomenon of entanglement sudden death appears more early in the system for the case of squeezed vacuum state than in the case of squeezed thermal state. Thereafter, it is also observed that non-commutativity effects lead to an increasing of entanglement of initially entangled quantum states, and reduce the separability in the open quantum system. It turns out that a separable state in the usual commutative quantum mechanics might be entangled in non-commutative extension.

On the noncommutative Bondal–Orlov conjecture for some toric varieties

We show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of “weakly symmetric” unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that all toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne–Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern–Leistner and Sam which showed that such GIT quotient stacks are a geometric incarnation of a family of specific toric NCCRs constructed earlier by the authors.

Noncommutative effective loop quantum cosmology: Inclusion of a potential term

We construct and study a simple noncommutative scheme (theta-deformation) for the effective Loop Quantum Cosmology of the flat Friedmann-Lema\^itre-Robertson-Walker model in the presence of a homogeneous scalar field $\phi$ with a potential $\mathcal{V}(\phi)=\frac{1}{2}m^2\phi^2$. We first conduct a simple analysis from the corresponding Hamilton equations of motion considering a generic term $\mathcal V(\phi)$. It is observed that the characteristic Big Bounce of Loop Quantum Cosmology is preserved under such noncommutative extension. When specializing to the quadratic case, numerical solutions to the corresponding Hamilton equations exhibiting an early inflationary epoch with a sufficiently large number of e-foldings are found. It is concluded that, in this noncommutative setup, solutions exist which are in the overall compatible with the early universe predicted by standard (effective) Loop Quantum Cosmology (i.e. a bouncing and inflationary early universe). The issue of the genericness of a sufficiently long inflationary period on the space of solutions in this noncommutative construct remains to be addressed.

Convolution algebra for extended Feller convolution

We apply the recently introduced framework of admissible homomorphisms in the form of a convolution algebra of $$\mathbb{C}^2$$ -valued admissible homomorphisms to handle two-dimensional uni-directional homogeneous stochastic kernels. The algebra product is a non-commutative extension of the Feller convolution needed for an adequate operator representation of such kernels: a pair of homogeneous transition functions uni-directionally intertwined by the extended Chapman–Kolmogorov equation is a convolution empathy; the associated Fokker–Planck equations are re-written as an implicit Cauchy equation expressed in terms of admissible homomorphisms. The conditions of solvability of such implicit evolution equations follow from the consideration of generators of a convolution empathy.

Bogolyubov Endomorphisms and Non-Commutative Hyperbolic Functions

We classify Bogolyubov endomorphisms of the CCR algebra over a general test function space. This classification requires a non-commutative extension of the usual hyperbolic functions. We consider semi-groups of such endomorphisms and characterize their generators under the assumption of norm continuity.

Ergodic properties of the Anzai skew-product for the non-commutative torus

We provide a systematic study of a non-commutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the non-commutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-product. As an application, for a uniquely ergodic Anzai skew-product $\unicode[STIX]{x1D6F7}$ on the non-commutative $2$-torus $\mathbb{A}_{\unicode[STIX]{x1D6FC}}$, $\unicode[STIX]{x1D6FC}\in \mathbb{R}$, we investigate the pointwise limit, $\lim _{n\rightarrow +\infty }(1/n)\sum _{k=0}^{n-1}\unicode[STIX]{x1D706}^{-k}\unicode[STIX]{x1D6F7}^{k}(x)$, for $x\in \mathbb{A}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D706}$ a point in the unit circle, and show that there are examples for which the limit does not exist, even in the weak topology.

Some new categorical invariants

In this paper we introduce new categorical notions and give many examples. In an earlier paper we proved that the Bridgeland stability space on the derived category of representations of K(l), the l-Kronecker quiver, is biholomorphic to {{mathbb {C}}} times {mathcal {H}} for lge 3. In the present paper we define a new notion of norm, which distinguishes {D^b(K(l)) }_{lge 2}. More precisely, to a triangulated category {mathcal {T}} which has property of a phase gap we attach a non-negative real number left| {mathcal {T}}right| ^{varepsilon }. Natural assumptions on a SOD {mathcal {T}} =langle {mathcal {T}}_1,{mathcal {T}}_2rangle imply left| langle {mathcal {T}}_1,{mathcal {T}}_2rangle right| ^{varepsilon }le {mathrm{min}}{left| {mathcal {T}}_1 right| ^{varepsilon },left| {mathcal {T}}_2right| ^{varepsilon } }. Using the norm we define a topology on the set of equivalence classes of proper triangulated categories with a phase gap, such that the set of discrete derived categories is a discrete subset, whereas the rationality of a smooth surface S ensures that [D^b(point)] in mathrm{Cl}([D^b(S)]). Categories in a neighborhood of D^b(K(l)) have the property that D^b(K(l)) is embedded in each of them. We view such embeddings as non-commutative curves in the ambient category and introduce categorical invariants based on counting them. Examples show that the idea of non-commutative curve-counting opens directions to new categorical structures and connections to number theory and classical geometry. We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi–Yau curve-counting, where the entities we count are a Calabi–Yau modification of D^b(K(l)). In the end we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm, introduced here, playing a role similar to the classical notion of degree of an extension in Galois theory.

The Hirota–Satsuma coupled KdV hierarchy on noncommutative phase-space

In the present contribution, the Hirota-Satsuma coupled KdV hierarchy on noncommutative phase-space is investigated using the noncommutative extension of Lax pair generating technique. In particular, the explicit representation of its associated Lax pair operators is constructed. It is shown that the obtained results for phase-space noncommutativity case reduce back to the standard commutative case when θ goes to ½.

Noncommutativity in Effective Loop Quantum Cosmology

We construct a noncommutative extension of the Loop Quantum Cosmology effective scheme for the flat FLRW model with a free scalar field via a theta deformation. Firstly, a deformation is implemented in the configuration sector, among the holonomy variable and the matter degree of freedom. We show that this type of noncommutativity retains, to some degree, key features of the Loop Quantum Cosmology paradigm for a free field. Secondly, a deformation is implemented in the momentum sector, among the momentum associated with the holonomy variable and the momentum associated with the matter field. We show that in this latter case the scalar field energy density is the same as the one in standard Loop Quantum Cosmology.

AdS-CFT correspondence in noncommutative background, related thermodynamics and holographic superconductor in magnetic field

In this work, we formulate a non-commutative (NC) extension of AdS-CFT correspondence that is manifested in the modification of behavior of a holographic superconductor. The noncommutativity is introduced in the model through the NC corrected AdS charged black hole metric developed by Nicolini, Smailagic and Spallucci. First of all we discuss thermodynamic properties of this black hole in Euclidean formalism. In particular, we compute trace of the boundary energy-momentum tensor which, as expected, is non-zero due to the NC scale introduced in the model. Our findings indicate that the non-commutative effects tend to work against the black hole hair formation. This, in turn, has an adverse effect on the holographic superconductor by making the superconducting phase more fragile. This is reflected in the reduced value of the critical magnetic field and critical temperature. Finally, we comment on a qualitative agreement between our (holographic superconductor) result and that obtained for a conventional superconductor in NC space in a purely condensed matter scenario. In both cases noncommutativity tends to oppose the superconducting phase.

The $$\lambda $$ λ -Function in the Space of Trace Class Operators

Let $$C_1(H)$$ denote the space of all trace class operators on an arbitrary complex Hilbert space H. We prove that $$C_1(H)$$ satisfies the $$\lambda $$ -property, and we determine the form of the $$\lambda $$ -function of Aron and Lohman on the closed unit ball of $$C_1(H)$$ by showing that $$\begin{aligned} \lambda (a) = \frac{1 - \Vert a\Vert _1 + 2 \Vert a\Vert _{\infty }}{2}, \end{aligned}$$ for every a in $${C_1(H)}$$ with $$\Vert a\Vert _1 \le 1$$ . This is a non-commutative extension of the formula established by Aron and Lohman for $$\ell _1$$ .

Fractals and chaos related to Ising-Onsager lattices. Ternary approach versus binary approach

In connection with several attempts to extend Onsager’s (1944) crystal statistics to three dimensions keeping the spirit of the original Ising’s (1925) model, we suggest to involve binary and ternary extension of triangles in connection with fractals and chaos, following the ideas of Kikuchi (1951). We apply ternary Clifford algebras in addition to the usual (binary) Clifford algebras and identify two ternaries with three ternaries with the help of noncommutative Galois extension. In some sense, the paper continues the ideas of Ławrynowicz, Marchiafava and Nowak-Kȩpczyk (2006), and Kerner and Suzuki (2012) published in this journal.

Noncommutative gravity with self-dual variables

We build a noncommutative extension of Palatini–Holst theory on a twist-deformed spacetime, generalizing a model that has been previously proposed by Aschieri and Castellani. The twist deformation entails an enlargement of the gauge group, and leads to the introduction of new gravitational degrees of freedom. In particular, the tetrad degrees of freedom must be doubled, thus leading to a bitetrad theory of gravity. The model is shown to exhibit new duality symmetries. The introduction of the Holst term leads to a dramatic simplification of the dynamics, which is achieved when the Barbero–Immirzi parameter takes the value , corresponding to a self-dual action. We study in detail the commutative limit of the model, focusing in particular on the role of torsion and non-metricity. The effects of spacetime noncommutativity are taken into account perturbatively, and are computed explicitly in a simple example. Connections with bimetric theories and the role of local conformal invariance in the commutative limit are also explored.