Dimensional Lie Algebra Research Articles

The algebraic classification of low-dimensional nilpotent n-Lie algebras

In this paper, we study the low-dimensional nilpotent [Formula: see text]-Lie algebras and classify nilpotent [Formula: see text]-Lie algebras of dimension [Formula: see text] and nilpotent [Formula: see text]-Lie algebras of class two with dimension [Formula: see text].

Strong Kähler with torsion solvable lie algebras with codimension 2 nilradical

AbstractIn this paper, we study strong Kähler with torsion (SKT) and generalized Kähler structures on solvable Lie algebras with (not necessarily abelian) codimension 2 nilradical. We treat separately the case of ‐invariant nilradical and non‐‐invariant nilradical. A classification of such SKT Lie algebras in dimension 6 is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized Kähler compact solvmanifolds.

Model-theoretic properties of nilpotent groups and Lie algebras

We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent p studied by Baudisch is 2-dependent and NSOP1. We prove that the class of c-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for 2<c, the generic c-nilpotent Lie algebra over Fp is strictly NSOP4 and c-dependent. Via the Lazard correspondence, we obtain the same result for c-nilpotent groups of exponent p, for an odd prime p>c.

Optimal systems of Lie subalgebras: A computational approach

Lie groups of symmetries of differential equations constitute a fundamental tool for constructing group-invariant solutions. The number of subgroups is potentially infinite and so the number of invariant solutions; thus, it is crucial to obtain a classification of subgroups in order to have an optimal system of inequivalent solutions from which all other solutions can be derived by action of the group itself. Since Lie groups are intimately connected to Lie algebras, a classification of inequivalent subgroups induces a classification of inequivalent Lie subalgebras, and vice versa. A general method for classifying the Lie subalgebras of a finite–dimensional Lie algebra uses inner automorphisms that are obtained by exponentiating the adjoint groups. In this paper, we present an effective algorithm able to automatically determine optimal systems of Lie subalgebras of a generic finite–dimensional Lie algebra abstractly assigned by means of its structure constants, or realized in terms of matrices or vector fields, or defined by a basis and the set of non-zero Lie brackets. The algorithm is implemented in the computer algebra system Wolfram Mathematica™; some meaningful and non-trivial examples are considered.

Lie semisimple algebras of derivations and varieties of PI-algebras with almost polynomial growth

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras U T 2 ( W λ ) UT_2(W_\lambda ) or E n d ( W μ ) End(W_\mu ) for some integral dominant weight λ , μ \lambda ,\mu with μ ≠ 0 \mu \neq 0 . In the special case L = s l 2 L=\mathfrak {sl}_2 we prove that this is a sufficient condition too.

The Core of Inner ideal of Real Four Dimensional Lie Algebra With One Dimensional Derived

Suppose that V is any inner ideal of L. The core of V is an inner ideal of L with special requirement. In this paper we prove. If L is a 4-dimension Lie algebra a with 1-dimensional derived , then the core of every inner ideal of L is zero. Moreover L containing a sandwich elements if L' not subset of Z and every element in L is sandwich if L' subset of Z. .

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 .

Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians

Based oh the properties of Lie algebras, in this work we develop a general framework to linearize the von Neumann equation rendering it in a suitable form for quantum simulations. Departing from the conventional method of expanding the density matrix in the Liouville space formed by matrices unit we express the von Neumann equation in terms of Pauli strings. This provides several advantages related to the quantum tomography of the density matrix and the formulation of the unitary gates that generate the time evolution. The use of Pauli strings facilitates the quantum tomography of the density matrix whose elements are purely real. As for any other basis of Hermitian matrices, this eliminates the need to calculate the phase of the complex entries of the density matrix. This approach also enables to express the evolution operator as a sequence of commuting Hamiltonian gates of Pauli strings that can readily be synthetized using Clifford gates. Additionally, the fact that these gates commute with each other along with the unique properties of the algebra formed by Pauli strings allows to avoid the use of Trotterization hence considerably reducing the circuit depth. The algorithm is demonstrated for three Hamiltonians using the IBM noisy quantum circuit simulator.

Automorphisms of projective structures

We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of foliated projective structures having positive dimensional Lie algebra of projective vector fields.

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.

Classification of flat Lorentzian nilpotent Lie algebras

AbstractWe give a complete classification of flat Lorentzian nilpotent Lie algebras, this is to say of pseudo‐Euclidean Lie algebras associated to nilpotent Lie groups endowed with a left‐invariant Lorentzian metric of vanishing curvature. We prove that every such a Lie algebra is a direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand and show that, if denotes the ‐dimensional Heisenberg Lie algebra, then the only non‐abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are and the semidirect products and , defined by some particular derivations . In all those cases we also find the equivalence classes of flat Lorentzian products.

Symmetry algebra classification of scalar n$$ n $$th‐order ordinary differential equations

We obtain a complete classification of scalar th‐order ordinary differential equations for all subalgebras of vector fields in the real plane. While softwares like Maple can compute invariants of a given order, our results are for a general . The cases are well‐known in the literature. Further, it is known that there are three types of th‐order equations depending upon the point symmetry algebra they possess, namely, first‐order equations which admit an infinite dimensional Lie algebra of point symmetries, second‐order equations possessing the maximum 8‐point symmetries, and higher‐order, , admitting the maximum dimensional point symmetry algebra. We show that scalar th‐order equations for do not admit maximally an dimensional real Lie algebra of point symmetries. Moreover, we prove that for , equations can admit two types of dimensional real Lie algebra of point symmetries: one type resulting in nonlinear equations which are not linearizable via a point transformation and the second type yielding linearizable (via point transformation) equations. Furthermore, we present the types of maximal real dimensional and higher than ‐dimensional point symmetry algebras admissible for equations of order and their canonical forms. The types of lower‐dimensional point symmetry algebras which can be admitted are shown, and the equations are constructible as well. We state the relevant results in tabular form and in theorems.

Symplectic Form yang Berkaitan Dengan Satu-form Suatu Aljabar Lie Berdimensi Rendah

In this paper, we study symplectic form on low dimensional real Lie algebra. A symplectic form is very important in classifying of Lie algebra types. Based on their dimension and certain conditions, there are two types of Lie algebras. A lie algebra with odd dimension endowed with one-form such that is called a contact Lie algebra, while a Lie algebra whose dimension is even and it is endowed with zero index is called a Frobenius Lie algebra. The research aimed to give explicit formula of a symplectic form of low dimensional contact Lie algebras and Frobenius Lie algebras. We established that a one-form associated to simplectic form determine a type of a Lie algebra whether a contact or a Frobenius Lie algebras.To clearer the main results, we give some examples of one-form and symplectic form of Frobenius and contact Lie algebras.

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.

Klasifikasi Aljabar Lie Forbenius-Quasi Dari Aljabar Lie Filiform Berdimensi ≤ 5

In this research, we studied quasi-Frobenius Lie algebras and filiform Lie algebras of dimensions ≤ 5 over real field. The primary objective of this research is to classify the classification of filiform Lie algebras of dimensions ≤ 5 into quasi-Frobenius Lie algebras. The method employed in this research involves constructing a skew-symmetric 2-form in real Lie algebra, which also a nondegenerate 2-cocycle. The outcomes of this research reveal that there exists a class of filiform Lie algebras of dimensions $\le 5$ that can be classified as a quasi-Frobenius real Lie algebra. Furthermore, this research can be developed to classify higher dimensional filiform Lie algebras as quasi-Frobenius real Lie algebras.

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.

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.

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.

Homological properties of metabelian restricted Lie algebras

We classify when a metabelian restricted Lie algebra over a field k is finitely presented independently from the type of the extension or the field k. We classify when a split extension abelian-by-abelian restricted Lie algebra is of homological type FPm.