Dimensional Lie Algebra Research Articles

The explicit power series solution formation and computationof Lie point infinitesimals generators: Lie symmetry approach

This work scrutinizes the well-known nonlinear non-classical Sobolev-type wave model which addresses the fluid flow via fractured rock, thermodynamics and many other fields of modern sciences. The nonlinear non-classical Sobolev-type wave model provides a more comprehensive and accurate description of wave phenomena in a wide range of fields. By incorporating both nonlinearity and the complexities of dispersive waves, these models enhance our understanding of natural phenomena and enable more precise predictions and applications in various scientific and engineering disciplines. Therefore, this study is investigating it. Prior to this study, no previous research has performed Lie symmetry analysis and achieved invariant solutions of this kind. The symmetry generators are taking into account the Lie invariance criteria. The suggested approach produces the three dimensional Lie algebra, where translation symmetries in space and time are associated with mass conservation and conservation of energy, respectively and the other symmetries are scaling or dilation. The nonlinear non-classical Sobolev-type wave partial differential equation is transformed into a system of highly nonlinear ordinary differential equations by employing appropriate similarity transformations through using Lie group methodology. The power series technique is used to generate exact wave solutions because the inverse scattering transform cannot solve the Cauchy problem for this equation. The graphical behaviour of certain solutions is demonstrated in 3-D and 2-D for particular quantities of the physical factors in the investigated equation.

Dirac cohomology for the BGG category [formula omitted

We study Dirac cohomology HDg,h(M) for modules belonging to category O of a finite dimensional complex semisimple Lie algebra. We start by studying the generalized infinitesimal character decomposition of M⊗S, with S being a spin module of h⊥. As a consequence, “Vogan’s conjecture” holds, and we prove a nonvanishing result for HDg,h(M) while we show that in the case of a Hermitian symmetric pair (g,k) and an irreducible unitary module M∈O, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in M. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for M∈O.

Lie symmetry analysis for the Cargo–Leroux model with isentropic perturbation pressure equation of state

In this article, the well-known Cargo–Leroux model with isentropic perturbation equation of state is analyzed using the Lie symmetry method. By using invariant conditions of system of partial differential equations, six dimensional Lie algebra is obtained. The optimal system for system of partial differential equations is constructed using adjoint representation and the invariants of associated Lie algebras of the system. Further, with the help of one-dimensional optimal system invariant solutions are constructed. Also, physically significant solutions such as traveling wave solutions, specifically the kink-type solitons and peakon-type solitons are obtained by using traveling wave transformations and all the solutions are graphically demonstrated. Finally, the hyperbolic nature of system of partial differential equations is examined by studying the evolutionary behavior of a discontinuity wave.

The computation of Lie point symmetry generators, modulational instability, classification of conserved quantities, and explicit power series solutions of the coupled system

The well-known Chaffee–Infante reaction hierarchy is examined in this article along with its reaction–diffusion coupling. It has numerous variety of applications in modern sciences, such as electromagnetic wave fields, fluid dynamics, high-energy physics, ion-acoustic waves in plasma physics, coastal engineering, and optical fibres. The physical processes of mass transfer and particle diffusion might be expressed in this way. The Lie invariance criteria is taken into consideration while we determine the symmetry generators. The suggested approach produces the six dimensional Lie algebra, where translation symmetries in space and time are associated to mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. Additionally, similarity reductions are performed, and the optimal system of the sub-algebra should be quantified. There are an enormous number of exact solutions can construct for the traveling waves when the governing system is transformed into ordinary differential equations using the similarity transformation technique. The power series approach is also utilized for ordinary differential equations to obtain closed-form analytical solutions for the proposed diffusive coupled system. The stability of the model under the limitations is ensured by the modulation instability analysis. The reaction diffusion hierarchy’s conserved vectors are calculated using multiplier methods using Lie Backlund symmetries. The acquired results are presented graphically in 2-D and 3-D to demonstrate the wave propagation behavior.

Teleparallel Robertson-Walker Geometries and Applications

In teleparallel geometries, the coframe and corresponding spin connection are the principal geometric objects and, consequently, the appropriate definition of a symmetry is that of an affine symmetry. The set of invariant coframes and their corresponding spin connections that respect the full six dimensional Lie algebra of Robertson–Walker affine symmetries are displayed and discussed. We will refer to such geometries as teleparallel Robertson–Walker (TRW) geometries, where the corresponding derived metric is of Robertson–Walker form and is characterized by the parameter k=(−1,0,1). The field equations are explicitly presented for the F(T) class of teleparallel TRW spacetimes. We are primarily interested in investigating the k≠0 TRW models. After first studying the k=0 models and, in particular, writing their governing field equations in an appropriate form, we then study their late time stability with respect to perturbations in k in both the cases of a vanishing and non-vanishing effective cosmological constant term. As an illustration, we consider both quadratic F(T) theories and power-law solutions.

Locally conformal SKT almost abelian Lie algebras

A locally conformal SKT (shortly LCSKT) structure is a Hermitian structure (J,g) whose Bismut torsion 3-form H satisfies the condition dH=α∧H, for some closed non-zero 1-form α. This condition was introduced as a generalization of the SKT (or pluriclosed) condition dH=0. In this paper, we characterize the almost abelian Lie algebras admitting a Hermitian structure (J,g) such that dH=α∧H, for some closed 1-form α. As an application we classify LCSKT almost abelian Lie algebras in dimension 6. Finally, we also study on almost abelian Lie algebras the compatibility between the LCSKT condition and other types of Hermitian structures.

Restricted modules and associated vertex algebras of extended Heisenberg-Virasoro algebra

In this paper, a family of infinite dimensional Lie algebras L˜ is introduced and investigated, called the extended Heisenberg-Virasoro algebra, denoted by L˜. These Lie algebras are related to the N=2 superconformal algebra and the Bershadsky-Polyakov algebra. We study restricted modules and associated vertex algebras of the Lie algebra L˜. More precisely, we construct its associated vertex (operator) algebras VL˜(ℓ123,0), and show that the category of vertex algebra VL˜(ℓ123,0)-modules is equivalent to the category of restricted L˜-modules of level ℓ123. Then we give uniform constructions of simple restricted L˜-modules. Also, we present several equivalent characterizations of simple restricted modules over L˜.

On Quasi-para-Sasakian Structures on 5-dimensions

In this study, we investigate the existence of quasi-para-Sasakian structures on five dimensional nilpotent
 Lie algebras. There are six non-abelian nilpotent Lie algebras. We show that quasi-para-Sasakian structures exist only on one of these algebras. Quasi-para-Sasakian structures correspond to the class G_5+G_8 in the classification of almost paracontact metric structures. We show that a quasi-para-Sasakian structure on a five dimensional nilpotent Lie algebra is either in G_5 or G_8.

Nilpotent Lie algebras with two centralizer dimensions over a finite field

A result of Barnea and Isaacs states that if L is a finite dimensional nilpotent Lie algebra with exactly two distinct centralizer dimensions, then nilpotency class of L is either 2 or 3. In this article, we classify all such finite dimensional 3-step nilpotent Lie algebras over a finite field.

О 7-мерных алгебрах голоморфных векторных полей в $ \Bbb C^4 $, имеющих 5-мерный абелев идеал

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in $ \Bbb C^4 $ we study in this article the 7-dimensional orbits of real Lie algebras in this space. By the well-known Morozov theorem, any nilpotent 7-dimensional Lie algebra has at least a 4-dimensional Abelian ideal. The article considers nilpotent indecomposable 7-dimensional Lie algebras containing a 5-dimensional Abelian ideal. It is proved that in the space $ \Bbb C^4 $ all the orbits of such algebras are Levi degenerate. This statement covers 73 algebras from the complete list of 149 indecomposable 7-dimensional nilpotent Lie algebras.

On toral posets and contact Lie algebras

A (2k+1)-dimensional Lie algebra is called contact if it admits a one-form φ such that φ∧(dφ)k≠0. Here, we extend recent work to describe a combinatorial procedure for generating contact, type-A Lie poset algebras whose associated posets have chains of arbitrary cardinality, and we conjecture that our construction leads to a complete characterization.

Varieties of subalgebras and endotrivial modules

Let (g,[p]) be a finite dimensional restricted Lie algebra over a perfect field k of characteristic p≥3. By combining methods from recent work of Benson-Carlson [4] with those of [10,16] we obtain a description of the endotrivial (g,[p])-modules in case the underlying Lie algebra g is supersolvable. For such g and algebraically closed k, this yields a classification of the indecomposable (g,[p])-modules of constant Jordan type with one non-projective block.

Poisson cohomology of 3D Lie algebras

We compute the Poisson cohomology associated with several three dimensional Lie algebras. Together with existing results and the classification of three dimensional Lie algebras, this provides the Poisson cohomology of all linear Poisson structures in dimension 3.

Braided Hopf Algebras and Gauge Transformations II: *-Structures and Examples

We consider noncommutative principal bundles which are equivariant under a triangular Hopf algebra. We present explicit examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres. The braiding of these algebras is implemented by the triangular structure of the symmetry Hopf algebra. We present a systematic analysis of compatible *-structures, encompassing the quasitriangular case.

Invariants of Weyl Group Action and q-characters of Quantum Affine Algebras

Let W be the Weyl group corresponding to a finite dimensional simple Lie algebra $$\mathfrak {g}$$ of rank $$\ell $$ and let $$m>1$$ be an integer. In Inoue (Lett. Math. Phys. 111(1):32, 2021), by applying cluster mutations, a W-action on $$\mathcal {Y}_m$$ was constructed. Here $$\mathcal {Y}_m$$ is the rational function field on $$cm\ell $$ commuting variables, where $$c \in \{ 1, 2, 3 \}$$ depends on $$\mathfrak {g}$$ . This was motivated by the q-character map $$\chi _q$$ of the category of finite dimensional representations of quantum affine algebra $$U_q(\hat{\mathfrak {g}})$$ . We showed in Inoue (Lett. Math. Phys. 111(1):32, 2021) that when q is a root of unity, $$\textrm{Im} \chi _q$$ is a subring of the W-invariant subfield $$\mathcal {Y}_m^W$$ of $$\mathcal {Y}_m$$ . In this paper, we give more detailed study on $$\mathcal {Y}_m^W$$ ; for each reflection $$r_i \in W$$ associated to the ith simple root, we describe the $$r_i$$ -invariant subfield $$\mathcal {Y}_m^{r_i}$$ of $$\mathcal {Y}_m$$ .

On Orbits of 4-Dimensional Representations of 3-Dimensional Solvable Lie Algebras

On Orbits of 4-Dimensional Representations of 3-Dimensional Solvable Lie Algebras

Nilpotent Lie algebras of breadth type (0, 3)

For a natural number m, a Lie algebra L over a field k is said to be of breadth type if the co-dimension of the centralizer of every non-central element is m. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type over of odd characteristic up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, and . We also discuss 2-step nilpotent Camina Lie algebras.

Weight Multiplicities and Young Tableaux Through Affine Crystals

The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac–Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the ( ( spin ) ) rigid tableaux, and prove that they are equinumerous to the weight multiplicities of the highest weight modules under our consideration. These new classes of Young tableaux arise from crystal basis elements for dominant maximal weights of the integrable highest weight modules over affine Kac–Moody algebras. By applying combinatorics of tableaux such as the Robinson–Schensted algorithm and new insertion schemes, and using integrals over orthogonal groups, we reveal hidden structures in the sets of weight multiplicities and obtain explicit closed formulas for the weight multiplicities. In particular we show that some special families of weight multiplicities form the Pascal, Catalan, Motzkin, Riordan and Bessel triangles.

Finite generation of Lie algebras of Poisson brackets

Let L be a finite dimensional Lie algebra. Let U(L) and Pol(L) be the universal enveloping algebra of L and the related Poisson algebra. We find conditions for finite generation of Lie algebras [U(L),U(L)] and [Pol(L), Pol(L)].

Construction of color Lie algebras from homomorphisms of modules of Lie algebras

A close relation between categories of color Lie algebras and sets of pairs of homomorphisms of modules of Lie algebras is discussed, which is applicable to construct color Lie algebras from Lie algebras and their modules. It is also shown that such construction yields equivalent categories of color Lie algebras under an equivalent relation between symmetric bicharacters. As an application we construct and classify all simple color Lie algebras from the three dimensional simple complex Lie algebra and its finite-dimensional simple modules.