Lie Algebra Research Articles

On different approaches to integrable lattice models

Interaction-round the face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e. the models the Boltzmann weights of which satisfy the quantum Yang–Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras su(2)k and su(3)k are computed for fundamental and adjoint representations for some fixed levels k. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on su(3)k (for general k) and discuss how it can be used to find Boltzmann weights in terms of the quantum R^ matrix when the adjoint representation defines the admissibility conditions.

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.

GROUP OF SYMMETRIES FOR THE DYNAMICS SYSTEM OF EQUATIONS OF RAREFIED TWO-PHASE MEDIUM

The symmetries of the system of equations of a two-phase medium are found, where the first phase is gas and the second one is solid particles. The second phase is considered rarefied, it is expressed in the absence of pressure in the equations of motion of the second phase. The medium is assumed to be nonisothermal. Using the methods of group analysis, the Lie algebras of symmetries of the model under study are found in the one-dimensional and three-dimensional cases. The paper describes in detail the process of searching for symmetries in the case of equations of state for a perfect gas. Some partially invariant solutions are found for the one-domensional system.

Computational approach in obtaining analytic solutions of a generalized nonlinear breaking soliton equation with applications in engineering and physics

Higher-order nonlinear wave models have been a source of attraction to a huge number of researchers in recent times as a result of their significance in mathematical physics, other nonlinear sciences as well as engineering. In consequence, we outline in this paper the analytical studies entrenched on a generalized structure of a nonlinear breaking soliton equation with higher-order nonlinearity in four variables which have applications in science as well as engineering. Lie group theory is utilized to generate an 11-dimensional Lie algebra associated with the equation under consideration and in addition one parametric group of transformations related to the algebra is calculated. Besides, the technique is further invoked in performing reductions of the various subalgebras of the understudy model. Moreover, in conjunction with the theory, the direct integration technique is engaged to secure an analytic solution of the equation and as a result, a general analytic solution with regard to the second-kind elliptic-integral function is furnished. Moreover, we engage the novel simplest equation technique to gain more general solutions to the equation. In consequence, solitonic solutions comprising periodic, dark-bright, topological kink as well as singular solutions are achieved. We complemented that by exhibiting the dynamics of the secured solutions with the aid of graphical representations. In conclusion, we calculate conserved quantities associated with the aforementioned equation by invoking the well-celebrated classical Noether theorem technique.

Deformations of modified $r$-matrices and cohomologies of related algebraic structures

Modified r -matrices are solutions of the modified classical Yang–Baxter equation, introduced by Semenov-Tian-Shansky, and play important roles in mathematical physics. In this paper, first we introduce a cohomology theory for modified r -matrices. Then we study three kinds of deformations of modified r -matrices using the established cohomology theory, including algebraic deformations, geometric deformations and linear deformations. We give the differential graded Lie algebra that governs algebraic deformations of modified r -matrices. For geometric deformations, we prove the rigidity theorem and study when is a neighborhood of a modified r -matrix smooth in the space of all modified r -matrix structures. In the study of trivial linear deformations, we introduce the notion of a Nijenhuis element for a modified r -matrix. Finally, applications are given to study deformations of the complement of the diagonal Lie algebra and compatible Poisson structures.

Group Classification of the Unsteady Axisymmetric Boundary Layer Equation

Unsteady equations of flat and axisymmetric boundary layers are considered. For the unsteady axisymmetric boundary layer equation, the problem of group classification is solved. It is shown that the kernel of symmetry operators can be extended by no more than four-dimensional Lie algebra. The kernel of symmetry operators of the unsteady flat boundary layer equation is found and it is shown that it can be extended by no more than a five-dimensional Lie algebra. The non-existence of the unsteady analogue of the Stepanov–Mangler transformation is proved.

The kernels of powers of linear operator via Weyr characteristic

The adjoint of a matrix in the Lie algebra associated with a matrix algebra is a fundamental operator, which can be generalized to a more general operator φ AB : X → AX − XB by two matrices A and B. The kernel of the operator is very well-known and it can be found in Gantmacher's book. The formulas for the dimensions of the kernels of arbitrary powers of the operator φ AB were given in terms of the Segre characteristics of these two matrices by the second and third authors in this paper and their collaborators. This paper provides an alternative approach to this problem via the Weyr characteristic in a more essential method. We obtain formulas for the dimensions of the kernels of arbitrary powers of the operator in terms of the Weyr characteristics. Furthermore, the basis for kernel of each power of the operator is described explicitly. As a consequence, for arbitrary square matrices A and B over an algebraically closed field, the dimension of the kernel of each power of the operator φ A − λI , B for eigenvalues λ of φ AB can be viewed as a similarity invariant of the operator φ AB , so we characterize the operator within similarity, which should be of interest to a number of people (including physicists).

Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering

Abstract This research article analytically investigates a soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences and engineering. The soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences along with engineering, is examined with a view to securing various pertinent results of interest. For the first time, the conserved currents of an integrodifferential equation (especially those of higher dimensions) are calculated using a detailed optimal system of one-dimensional subalgebras. Infinitesimal generators of diverse structures ascribed to Lie point symmetries of the understudy model are first calculated via Lie group analysis technique. Additionally, we construct various commutations along Lie-adjoint representation tables connected to the nine-dimensional Lie algebra achieved. Further to that, detailed and comprehensive computation of the optimal system of one-dimensional subalgebras linked to the algebra is also unveiled for the under-investigated model. This, in consequence, engenders the calculation of abundant conserved currents for the soliton equation through Ibragimov’s conserved vector theorem by utilizing its formal Lagrangian. Later, the applications of our results are highlighted.

On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities

Lie algebra plays an important role in the study of singularity theory and other fields of sciences. Finding numerous invariants linked with isolated singularities has always been a primary interest in the field of classification theory of isolated singularities. Any Lie algebra that characterizes simple singularity produces a natural question. The study of properties such as to find the dimensions of newly defined algebra is a remarkable work. Hussain, Yau and Zuo have found a new class of Lie algebra Lk(V), k≥1, i.e., Der (Mk(V),Mk(V)) and proposed a conjecture over its dimension δk(V) for k≥0. Later, they proved it true for k up to k=1,2,3,4,5. In this work, the main concern is whether it is true for a higher value of k. According to this, we first calculate the dimension of Lie algebra Lk(V) for k=6 and then compute the upper estimate conjecture of fewnomial isolated singularities. Additionally, we also justify the inequality conjecture δk+1(V)<δk(V) for k=6. Our calculated results are innovative and serve as a new addition to the study of singularity theory.

The Lie point symmetry criteria and formation of exact analytical solutions for Kairat-II equation: Paul-Painlevé approach

The renowned non-linear Kairat-II equation, which expresses the surface curves, is examined meticulously in this work. There has never been a study before that discussed conserved quantities, sensitivity analysis, Hamiltonian function, Lie symmetry invariance criteria, and invariant solutions of Kairat-II equation The Lie invariance criteria are taken into consideration by the symmetry generators. The suggested technique leads to in a four-dimensional Lie algebra, where the translation point symmetries in space and time correlate with the conservation of mass and the energy, respectively, and remaining point symmetries are dilation and scaling. Firstly, we use symmetry reduction of Lie subalgebras to obtain closed-form invariant solutions. In specific reduction cases, we transform the Kairat-II equation into a spectrum of non-linear ordinary differential equations, which have the benefit of being able to offer an extensive selection of closed-form solitary wave solutions. For this equation, the Cauchy problem cannot be solved by the inverse scattering transform, therefore, traveling wave exact solutions are generated using the analytical Paul-Painlevé approach. In both two and three dimensions, the graphical behavior of specific solutions is presented for specific quantities of physical factors of examined equation. Utilizing conservation laws multipliers, the study culminates by determining an extensive set of local conservation laws for the non-linear Kairat-II equation that are applicable to arbitrary constant coefficients. Instead of focusing on the physical aspects of conservation laws, this study computed the conserved quantities using a mathematical perspective that can be used to identify potential symmetries of partial differential equations, guiding readers toward the solutions of partial differential equations. The existence of the Hamiltonian function is presented. The model’s sensitivity to various starting conditions is highlighted by the sensitivity analysis.

Comments on Parallel Curves in 3-Dimensional Lie Group G

In this study, firstly, basic concepts in 3-dimensional Euclidean space and basic information about curves are given and some special curves are examined. Then, basic information about Lie algebra and Lie group basic concepts and curves are given and special curves such as helix, involute-evolute, Bertrand, Mannheim, Smarandache, are defined. Secondly, inspired by these special curves examined in the Lie group, the definitions of the parallel curve in the vector direction, the parallel curve in the direction of the vector B and the parallel curve in the direction of the linear combination of the vectors B and N of a curve according to the Frenet frame are given and characterized. Some theorems and results are obtained by finding the Frenet apparatus of these characterized curves. Finally, the findings are examined in more specific circumstances and new results are found.

Isotopisms of nilpotent Leibniz algebras and Lie racks

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J 1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks.

Reduced dynamics and geometric optimal control of nonequilibrium thermodynamics: Gaussian case

This paper introduces a control-theoretic formulation of nonequilibrium thermodynamic systems with Gibbs states of Gaussian distributions, i.e., thermodynamic systems characterized by Gaussian probability densities. The distinct features of the paper are three-fold. First, the dynamics of a thermodynamic system is studied by transferring the original state variable from the non-Euclidean and nonlinear manifold state space to a Lie group, and further to the dual space of a Lie algebra, which is endowed with vector space structures. Consequently, the resulting reduced dynamics significantly reduces the high nonlinearity appearing in the original non-Euclidean state space of a thermal process. Second, the obtained equations of motion interestingly indicate that thermodynamics can be naturally viewed as a generalization of rigid body motions, and this bridges control theory, thermodynamics, information theory, and rigid body dynamics. Third, the optimality conditions for the energy-minimum optimal control problem of probability densities are derived via geometric Pontryagin’s principle by regarding the reduced dynamics as dynamical constraints, and a unified control algorithm is developed. Finally, the proposed approach is applied to three different scenarios including two benchmark examples to demonstrate the applicability and effectiveness. The purpose of the paper is to provide a deeper understanding of thermodynamics from a control perspective, and also to draw intrinsic connections between control theory, thermodynamics, information theory, and rigid body dynamics.

Correlated dynamics of immune network and sl(3, R) symmetry algebra

Abstract We observed the existence of periodic orbits in immune network under transitive solvable Lie algebra. In this article, we focus to develop condition of maximal Lie algebra for immune network model and use that condition to construct a vector field of symmetry to study nonlinear pathogen model. We used two methods to obtain analytical structure of solution, namely normal generator and differential invariant function. Numerical simulation of analytical structure exhibits correlated periodic pattern growth under spatiotemporal symmetry, which is similar to the linear dynamical simulation result. We used Lie algebraic method to understand correlation between growth pattern and symmetry of dynamical system. We employ idea of using one parameter point group of transformation of variables under which linear manifold is retained. In procedure, we present the method of deriving Lie point symmetries, the calculation of the first integral and the invariant solution for the ordinary differential equation (ODE). We show the connection between symmetries and differential invariant solutions of the ODE. The analytical structure of the solution exhibits periodic behavior around attractor in local domain, same behavior obtained through dynamical analysis.

Nilpotency and capability in multiplicative Lie algebras

This paper aims to introduce the concept of nilpotency and capability in multiplicative Lie algebras. Also, we see the existence of covers of a multiplicative Lie algebra and thoroughly examine their relationships with capable and perfect multiplicative Lie algebras.

Investigation of ID-derivations of finite-dimensional nilpotent Lie algebras

Let [Formula: see text] be a Lie algebra and let [Formula: see text] be the set of all derivations of [Formula: see text]. Then the derivation [Formula: see text] of [Formula: see text] is called an ID-derivation if [Formula: see text] for all [Formula: see text]. The set of all ID-derivations is denoted by [Formula: see text]. Let [Formula: see text] be the set of all central derivations and let [Formula: see text] and [Formula: see text] be, respectively, the set of all center derivations and ID-derivations that map [Formula: see text] to zero. In this paper, we verify relations between [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text]. Also, we prove that if [Formula: see text] is a nilpotent Lie algebra of class [Formula: see text], then [Formula: see text] is a nilpotent Lie algebra of class [Formula: see text].

Nonclassical symmetries, optimal classification, and dynamical behavior of similarity solutions of (3+1)-dimensional Burgers equation

This paper is devoted to the study of symmetry group classification, and optimal subalgebras of the three-dimensional Burgers equation, which describes nonlinear wave motion exhibits nonlinear effects and enhanced dispersion in the context of nonlinear dynamics. The equation undergoes a Painlevé analysis to assess its integrability properties to verify the possibility that it passes the Painlevé test, followed by the derivation of nonclassical symmetries. Utilizing the invariance property of Lie groups, desirable infinitesimal symmetries of Lie algebra have been outlined for the equation. Relying on the invariance characteristic of adjoint transformation along with the newly generated similarity variables, an intensive and systematically organized approach is employed to achieve a one-dimensional optimal system. The entire set of group invariant solutions for each of the associated subalgebras has been established. The dynamical behavior of derived exact solutions is examined using numerical simulations, and numerous intriguing occurrences are discovered, such as flat sheet, periodic wave, doubly soliton, multisoliton, bright-dark soliton, breather soliton, line soliton type, which offer valuable insights into the dynamics of the system and can predict the behavior of waves in real-world scenarios.

Bounds on cohomological support varieties

Over a local ring R R , the theory of cohomological support varieties attaches to any bounded complex M M of finitely generated R R -modules an algebraic variety V R ( M ) {\mathrm {V}}_R(M) that encodes homological properties of M M . We give lower bounds for the dimension of V R ( M ) {\mathrm {V}}_R(M) in terms of classical invariants of R R . In particular, when R R is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When M M has finite projective dimension, we also give an upper bound for dim ⁡ V R ( M ) \dim {\mathrm {V}}_R(M) in terms of the dimension of the radical of the homotopy Lie algebra of R R . This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of R R . Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.

Realization of Lie algebras and classifying spaces of crossed modules

Realization of Lie algebras and classifying spaces of crossed modules

Rings of finite Morley rank without the canonical base property

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over an algebraically closed field of characteristic [Formula: see text] give rise to triangular rings without the CBP. This also applies to Baudisch’s [Formula: see text]-step nilpotent Lie algebras, which yields the existence of a [Formula: see text]-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP.