Three-dimensional Lie Algebra Research Articles

The formation of invariant optical soliton structures to electric-signal in the telegraph lines on basis of the tunnel diode and chaos visualization, conserved quantities: Lie point symmetry approach

This study critically examines the nonlinear Lonngren wave model, a mathematical representation used to describe waves in shallow water environments such as beaches, lakes, and rivers, as well as electrical signals in semiconductor materials like tunnel diodes. The symmetry generators take into account the Lie invariance criteria. A three-dimensional Lie algebra can be obtained by using the suggested methodology. It links translation symmetries in space and time, respectively, with conservation of mass and energy. The third symmetry pertains to scaling or dilation. Recognizing the constraints of the inverse scattering transform in addressing the Cauchy problem for this equation, the investigation adopts new auxiliary equation approach to ordinary differential equations. This approach yields closed-form analytical solutions for the wave model. The obtained solutions include a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, smooth soliton, trigonometric solution, mixed singular solution, complex solitary and singular solution. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the total energy of the system in terms of its position and momentum. Chaos analysis is performed to describe periodic, quasi-periodic, and chaotic behaviours, and a sensitivity analysis highlights the model’s sensitivity to different wave numbers. Graphical representations of selected solutions are presented in both 3-D and 2-D, illustrating the behaviour under specific physical conditions. The study concludes by identifying a comprehensive set of local conservation laws for the nonlinear Lonngren wave equation, applicable to arbitrary constant coefficients, using conservation laws multipliers.

Lie group analysis, solitons, self-adjointness and conservation laws of the nonlinear elastic structural element equation

This study is based on the Lie group method for the nonlinear elastic structural element equation (ESE Equation). We obtain a three-dimensional Lie algebra. By utilizing this Lie algebra a four-dimensional optimal system is constructed. The governing ESE Equation is converted to nonlinear ordinary differential equations (ODEs) via symmetry reduction. We use a modified auxiliary equation (MAE) procedure to deal with nonlinear ODEs. These ODEs reveal the dynamics of the periodic and soliton solutions. We obtain soliton solutions through rational, trigonometric, and hyperbolic functions. Wolfram Mathematica simulations vividly illustrate the wave characteristics of the derived solutions, affirming their properties as singular periodic solutions, a singular solution, an optical dark soliton solution, and a singular soliton solution. We also obtain the local conservation laws by a new conservation theorem introduced by Ibragimov.

Classification of types A and A_+ from low dimensional standard and non-standard filiform Lie Algebras

In this paper, we study low-dimensional Filiform Lie algebras. Specifically, three-dimensional standard Filiform Lie algebras and five-dimensional non-standard Filiform Lie algebras. The classification method was given in the following stage. For given a low-dimensional Filiform Lie algebra, we compute its second centre. We showed that three-dimensional Filiform Lie algebra-called Heisenberg Lie algebra-is type 𝐴 and 𝐴+ as well. On the other hand, for 𝑛≥3, the standard Filiform Lie algebras are type 𝐴 but not type 𝐴+. In this case, we give a concrete example of case five-dimensional Heisenberg Lie algebra. Moreover, we proved that five-dimensional non-standard Filiform Lie algebra is type 𝐴 but not type 𝐴+. It is still an open problem to classify types 𝐴 and 𝐴+ for the general case of non-standard Filiform Lie algebra of dimension ≥6.

Using Vector-Product Loop Algebra to Generate Integrable Systems

A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equation, a generalized Schrödinger equation, a sine-Gordon equation, and a sinh-Gordon equation. Next, the well-known Tu scheme is generalized for generating isospectral integrable hierarchies and non-isospectral integrable hierarchies. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy presented in the paper is worded out. Finally, we further enlarge the three-dimensional loop algebra into a six-dimensional loop algebra so that a new isospectral integrable hierarchy which is a type of extended integrable model is produced whose bi-Hamiltonian structure is also derived from the vector-product trace identity. This new approach presented in the paper possesses extensive applications in the aspect of generating integrable hierarchies of evolution equations.

Rational equivalence for enveloping algebras of three-dimensional Lie algebras

We study from the point of view of rational equivalence the enveloping algebras of Lie algebras of dimension 3 whose derived Lie subalgebra is of dimension 2, over an algebraically closed base field in arbitrary characteristics.

Surjektifitas Pemetaan Eksponensial untuk Grup Lie Heisenberg yang Diperumum

The Heisenberg Lie Group is the most frequently used model for studying the representation theory of Lie groups. This Lie group is modular-noncompact and its Lie algebra is nilpotent. The elements of Heisenberg Lie group and algebra can be expressed in the form of matrices of size 3×3. Another specialty is also inherited by its three-dimensional Lie algebra and is called the Lie Heisenberg algebra. The Heisenberg Lie Group whose Lie Algebra is extended to the dimension 2n+1 is called the generalized Heisenberg Lie group and it is denoted by H whose Lie algebra is h_n. In this study, the surjectiveness of exponential mapping for H was studied with respect to h_n=⟨x ̅,y ̅,z ̅⟩ whose Lie bracket is given by [X_i,Y_i ]=Z. The purpose of this research is to prove the characterization of the Lie subgroup with respect to h_n. In this study, the results were obtained that if ⟨x ̅,y ̅ ⟩=:V⊆h_n a subspace and a set {e^(x_i ) e^(x_j ) ┤| x_i,x_j∈V }=:L⊆H then L=H and consequently Lie(L)≠V.

On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations

There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3≤r≤n), there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n−r)th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm.

Classification of a family of 4-dimensional anti-commutative algebras and their automorphisms

We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field K of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of K by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of automorphisms of the three-dimensional nilpotent Lie algebra by K.

Lie symmetry analysis for two-phase flow with mass transfer

Abstract We perform a complete symmetry classification for the hyperbolic system of partial differential equations, which describes a drift-flux two-phase flow in a one-dimensional pipe, with a mass-transfer term between the two different phases of the fluid. In addition, we consider the polytropic equation of states parameter and gravitational forces. For general values of the polytropic indices, we find that the fluid equations are invariant under the elements of a three-dimensional Lie algebra. However, additional Lie point symmetries follow for specific values of the polytropic indices. The one-dimensional systems are investigated in each case of the classification scheme, and the similarity transformations are calculated in order to reduce the fluid equations into a system of ordinary differential equations. Exact solutions are derived, while the reduced systems are studied numerically.

Involutions of sl(2,k) and non-split, three-dimensional simple Lie algebras

We give a process to construct non-split, three-dimensional simple Lie algebras from involutions of [Formula: see text], where [Formula: see text] is a field of characteristic not two. Up to equivalence, non-split three-dimensional simple Lie algebras obtained in this way are parametrized by a subgroup of the Brauer group of [Formula: see text] and are characterized by the fact that their Killing form represents [Formula: see text]. Over local and global fields we re-express this condition in terms of Hilbert and Legendre Symbols and give examples of three-dimensional simple Lie algebras which can and cannot be obtained by this construction over the field of rationals.

Examples of S—expansions of Lie Algebras

In this note, we give examples of S—expansions of Lie algebras of finite and infinite dimension. For the finite dimensional case, we expand all real three-dimensional Lie algebras. In the case of infinite dimension, we perform contractions obtaining new Lie algebras of infinite dimension.

Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

Generalized spinning particles on {mathcal {S}}^2 in accord with the Bianchi classification

Motivated by recent studies of superconformal mechanics extended by spin degrees of freedom, we construct minimally superintegrable models of generalized spinning particles on {mathcal {S}}^2, the internal degrees of freedom of which are represented by a 3-vector obeying the structure relations of a three-dimensional real Lie algebra. Extensions involving an external field of the Dirac monopole, or the motion on the group manifold of SU(2), or a scalar potential giving rise to two quadratic constants of the motion are discussed. A procedure how to build similar models, which rely upon real Lie algebras with dimensions d=4,5,6, is elucidated.

О симметриях обобщенного уравнения Кортевега–де Фриза

The Korteweg–de Vries equation is a third–order nonlinear partial differential equation that plays an important role in the theory of nonlinear waves, mainly of hydrodynamic origin. It was first obtained by Joseph Boussinesq in 1877, but a detailed analysis was already carried out by Diederik Korteweg and Gustav de Vries in 1895. The Korteweg–de Vries equation, its analogues and generalizations arise in mathematical models in a variety of subject areas. There are at least several thousand publications in which the Korteweg–de Vries equation is considered from one side or another, and the authors are aware of the impossibility of writing at least a simple list listing these publications. Nevertheless, it is possible to mention, for example, The paper considers some generalization of the Korteweg–de Vries equation obtained by introducing an arbitrary function with respect to which a group classification is performed. The admissible operators of the generalized Korteweg–de Vries equation, the basic algebra of admissible operators and possible cases of extension of the algebra of admissible operators for a coefficient function of a special kind are found. Tables of commutators of the obtained algebras are also calculated. At the same time, the fact of isomorphism of admissible algebras in two different cases of specification of the function-coefficient of the generalized Korteweg–de Vries equation is noted. It is shown that the maximum possible dimension of the algebra of admissible operators is equal to four and corresponds to the almost classical Korteweg–de Vries equation with a possible additional linear term. In all other cases, if the resulting algebra of admissible operators and extends, then to a three-dimensional Lie algebra.

A generalization on derivations of Lie algebras

<p style='text-indent:20px;'>We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.</p>

Deformations of Courant pairs and Poisson algebras

We study the deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base, which leads to describe the obstruction in extending a given deformation. We also discuss the construction of versal deformation of Courant pairs. As an application, we compute universal infinitesimal deformation of Poisson algebra structures on the three-dimensional complex Heisenberg Lie algebra. We compare the second deformation cohomology spaces of these Poisson algebra structures by considering them in the category of Leibniz pairs and Courant pairs, respectively.

Lie symmetries and singularity analysis for generalized shallow-water equations

Abstract We perform a complete study by using the theory of invariant point transformations and the singularity analysis for the generalized Camassa-Holm (CH) equation and the generalized Benjamin-Bono-Mahoney (BBM) equation. From the Lie theory we find that the two equations are invariant under the same three-dimensional Lie algebra which is the same Lie algebra admitted by the CH equation. We determine the one-dimensional optimal system for the admitted Lie symmetries and we perform a complete classification of the similarity solutions for the two equations of our study. The reduced equations are studied by using the point symmetries or the singularity analysis. Finally, the singularity analysis is directly applied on the partial differential equations from where we infer that the generalized equations of our study pass the singularity test and are integrable.

Deformation of Lie–Poisson algebras and chirality

Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian symmetric spectra: If λ is an eigenvalue of the linearized generator, −λ and λ¯ (hence, −λ¯) are also eigenvalues—the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Z. Yoshida et al., Phys. Lett. A 381, 2772–2777 (2017)]. To elucidate how chiral spectra are generated, we study the three-dimensional Lie–Poisson systems and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi’s list of all three-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra so(3), yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.

Levi flat CR structures on 3D Lie algebras

We completely classify Levi flat CR structures (that is, CR structures with vanishing Levi form) on three-dimensional real Lie algebras, in terms of their algebraic and almost contact properties.

Deformations of the three-dimensional Lie algebra sl(2)

Deformations of the three-dimensional Lie algebra sl(2)