Finite-dimensional Lie Algebra Research Articles

THE LEVI DECOMPOSITION OF THE LIE ALGEBRA M_2 (R)⋊gl_2 (R)

The idea of the Lie algebra is studied in this research. The decomposition between Levi sub-algebra and the radical can be used to define the finite dimensional Lie algebra. The Levi decomposition is the name for this type of decomposition. The goal of this study is to obtain a Levi decomposition of the Lie algebra of dimension 8. We compute its Levi sub-algebra and the radical of Lie algebra with respect to its basis to achieve this goal. We use literature studies on the Levi decomposition and Lie algebra in Dagli result to produce the radical and Levi sub-algebra. It has been shown that can be decomposed in the terms of the Levi sub-algebra and its radical. In this resulst, it has been given by direct computations and we obtained that the explicit formula of Levi decomposition of the affine Lie algebra whose basis is is written by with is is the Levi sub-algebra of .

Cohomological varieties associated to vertex operator algebras

Given a vertex operator algebra V, one can attach a graded Poisson algebra called the C2-algebra R(V). The associate Poisson scheme provides an important invariant for V and has been studied by Arakawa as the associated variety. In this article, we define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties and then construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational C2-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. In this paper, we study R(V) as a commutative algebra only and do not use the property of its Poisson structure, which is expected to provide more refined invariants. The goal of this work is to study the cohomological supports of modules for vertex algebras as the cohomological support varieties for finite groups and restricted 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].

A computational approach to almost-inner derivations

We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.

A study of Bianchi type I spacetime according to their Ricci collineations

Rici collineations (RCs) have been used in this research to study the locally rotationally symmetric (LRS) Bianchi type I spacetimes. To accomplish our objectives, the RC equations are typically integrated for both situations of the Ricci tensor, degenerate and non-degenerate. Throughout this work, a number of situations occur that provide various finite and infinite dimensional Lie algebras of RCs.

Classification of simple smooth modules over the Heisenberg–Virasoro algebra

In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra ${\mathfrak {D}}$ , and simple smooth modules over the twisted Heisenberg–Virasoro algebra $\bar {\mathfrak {D}}$ with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak {D}}$ . A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg–Virasoro vertex algebras. We also present a few examples of simple smooth ${\mathfrak {D}}$ -modules and $\bar {\mathfrak {D}}$ -modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak {D}$ and $\bar {\mathfrak {D}}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.

Transposed Poisson structures on solvable and perfect Lie algebras

We described all transposed Poisson algebra structures on oscillator Lie algebras, i.e. on one-dimensional solvable extensions of the -dimensional Heisenberg algebra; on solvable Lie algebras with naturally graded filiform nilpotent radical; on -dimensional solvable extensions of the -dimensional Heisenberg algebra; and on n-dimensional solvable extensions of the n-dimensional algebra with trivial multiplication. We also answered one question on transposed Poisson algebras early posted in a paper by Beites, Ferreira and Kaygorodov. Namely, we found that the semidirect product of and irreducible module gives a finite-dimensional Lie algebra with non-trivial -derivations, but without non-trivial transposed Poisson structures.

Bounds for the dimension of the Schur multiplier of finite dimensional nilpotent Lie algebras

Let L be an n-dimensional nilpotent Lie algebra of nilpotency class c≥2 over an arbitrary field F with dim⁡γ2(L)=m and dim(Z(L)Z(L)∩γ2(L))=t, where Z(L) and γ2(L) denote the center ideal and the derived subalgebra of L, respectively. The purpose of this work is to obtain a new bound on the dimension of the Schur multiplier for L, which is given in terms of n,m,c, and t. In the virtue of an old bound due to Niroomand and Russo, namely 12(n−m−1)(n+m−2)+1, we find a finite dimensional nilpotent Lie algebra L of nilpotency class 3 over a field F of characteristic 3 that attains this bound. We then present a complete list of all finite dimensional nilpotent Lie algebras of nilpotency class at least 2 that reach Niroomand–Russo's Formula. Finally, we give the structure of all finite dimensional nilpotent Lie algebras that attain some recent generalizations of Niroomand–Russo's Formula.

Realization of arbitrary Lie algebras by automorphisms of $\mathrm{CR}$ manifolds and symmetries of differential equations

For any finite-dimensional real Lie algebra $\mathfrak{h}$, we construct a germ of a real analytic hypersurface in complex space such that its Lie algebra of infinitesimal holomorphic automorphisms is isomorphic to $\mathfrak{h}$. For any $\mathfrak{h}$, we also construct a system of partial differential equations whose Lie algebra of symmetries is isomorphic to the complexification of the algebra $\mathfrak{h}$.

U0-free quantum group representations

Let g be a finite dimensional complex simple Lie algebra, and let U=Uq(g) be its quantized enveloping algebra with a triangular decomposition U=U−U0U+. We classify all U-module structures on U0 with the regular action of U0 on itself, which are shown to have direct connections with the q-Weyl algebra and the bounded weight representations. We obtain that the necessary and sufficient condition for the existence of such modules is that U has to be of type An(n≥1), Bn(n≥2) or Cn(n≥3). Moreover, we study their module structures and associated weight modules for each type.

Geometry-Preserving Numerical Methods for Physical Systems with Finite-Dimensional Lie Algebras

Geometry-Preserving Numerical Methods for Physical Systems with Finite-Dimensional Lie Algebras

Finite-Dimensional Nilpotent Lie Algebras with Central Derivation Lie Algebras of Minimal Possible

Finite-Dimensional Nilpotent Lie Algebras with Central Derivation Lie Algebras of Minimal Possible

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.

Biderivations of complete Lie algebras

The authors of this article intend to present some results obtained in the study of biderivations of complete Lie algebras. Firstly they present a matricial approach to do this, which was a useful and explanatory tool not only in the study of biderivations but also in the synthesis of these results. Then they study all biderivations of a Lie algebra [Formula: see text] with [Formula: see text] and [Formula: see text], called complete. Moreover, as an application of the previous result, they describe all biderivations of a semisimple Lie algebra (that are complete), extending a result obtained by Tang in [X. Tang, Biderivations of finite-dimensional complex simple Lie algebras, Linear Multilinear Algebra 66(2) (2018) 250–259] that describes all biderivations of a complex simple Lie algebra. And thirdly, results on symmetric and skew-symmetric biderivations are also presented.

Formula omitted] for cluster algebras from moduli spaces of G-local systems

For a finite-dimensional simple Lie algebra g admitting a non-trivial minuscule representation and a connected marked surface Σ with at least two marked points and no punctures, we prove that the cluster algebra Ag,Σ associated with the pair (g,Σ) coincides with the upper cluster algebra Ug,Σ. The proof is based on the fact that the function ring O(AG,Σ×) of the moduli space of decorated twisted G-local systems on Σ is generated by matrix coefficients of Wilson lines introduced in [23]. As an application, we prove that the Muller-type skein algebras Sg,Σ[∂−1][32,24,25] for g=sl2,sl3, or sp4 are isomorphic to the cluster algebras Ag,Σ.

Symmetries of the Energy–Momentum Tensor for Static Plane Symmetric Spacetimes

This article explores matter collineations (MCs) of static plane-symmetric spacetimes, considering the stress–energy tensor in its contravariant and mixed forms. We solve the MC equations in two cases: when the energy–momentum tensor is nondegenerate and degenerate. For the case of a degenerate energy–momentum tensor, we employ a direct integration technique to solve the MC equations, which leads to an infinite-dimensional Lie algebra. On the other hand, when considering the nondegenerate energy–momentum tensor, the contravariant form results in a finite-dimensional Lie algebra with dimensions of either 4 or 10. However, in the case of the mixed form of the energy–momentum tensor, the dimension of the Lie algebra is infinite. Moreover, the obtained MCs are compared with those already found for covariant stress–energy.

A new look at Lie algebras

We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. This pair allows to build a Lie bracket on a dual space to a linear space V. The Lie algebra obtained in this way is solvable. In particular, when F is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie–Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.

Equivalence of categories of simple Lie algebras in positive characteristic

In this paper we first study some properties of the finite-dimensional simple restricted Lie algebras. In the literature is found a one-to-one correspondence between them and finite-dimensional simple Lie algebras over a field of positive characteristic. Motivated by this fact, we give a one-to-one correspondence between their morphisms, which allow us to conclude that such categories are equivalent.

On Some Classes of Bases in Finite-Dimensional Lie Algebras

On Some Classes of Bases in Finite-Dimensional Lie Algebras

Contact Lie systems: theory and applications

A Lie system is a time-dependent system of differential equations describing the integral curves of a time-dependent vector field that can be considered as a curve in a finite-dimensional Lie algebra of vector fields V. We call V a Vessiot–Guldberg Lie algebra. We define and analyse contact Lie systems, namely Lie systems admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields relative to a contact manifold. We also study contact Lie systems of Liouville type, which are invariant relative to the flow of a Reeb vector field. Liouville theorems, contact Marsden–Weinstein reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Contact Lie systems on three-dimensional Lie groups with Vessiot–Guldberg Lie algebras of right-invariant vector fields and associated with left-invariant contact forms are classified. Our results are illustrated with examples having relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, quantum mechanical systems, etc. Finally, a Poisson coalgebra method to derive superposition rules for contact Lie systems of Liouville type is developed.