Filiform Lie Algebras Research Articles

The structure of filiform Lie algebra Q2m

The structure of filiform Lie algebra Q2m

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 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.

On Properties of Five-dimensional Nonstandard Filiform Lie algebra

In this paper, we study the five-dimensional nonstandard Filiform Lie algebra and their basis elements representations. The aim of this research is to determine the basis elements of five-dimensional nonstandard Filiform Lie algebras representation in the form of real matrices. The method used in this study is by following Ceballos, Núñez, and Tenorio’s work. The results of this study are five real matrices as the realization of the basis elements of the five-dimensional nonstandard Filiform Lie algebra. We also discuss some results relate to five-dimensional nonstandard Filiform Lie algebra’s properties. The five-dimensional nonstandard Filiform Lie algebra is always nilpotent. For further research, it can be extended to five classes of Filiform Lie algebra, both standard and nonstandard with six dimensions. Moreover, it can be computed their split torus such that their direct sums are Frobenius Lie algebras.

Q-Poincaré Inequalities on Carnot Groups with Filiform Type Lie Algebra

In this paper we prove (global) q- Poincaré inequalities for probability measures on nilpotent Lie groups with filiform Lie algebra of any length. The probability measures under consideration have a density with respect to the Haar measure given as a function of a suitable homogeneous norm.

Nilpotent algebras, implicit function theorem, and polynomial quasigroups

We prove an implicit function theorem for nilpotent (not necessarily associative or Lie) algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of the classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.

Extensions of solvable Lie algebras with naturally graded filiform nilradical

In this work, we consider extensions of solvable Lie algebras with naturally graded filiform nilradicals. Note that there exist two naturally graded filiform Lie algebras [Formula: see text] and [Formula: see text] We find all one-dimensional extensions of solvable Lie algebras with nilradical [Formula: see text]. We prove that there exists a unique non-split central extension of solvable Lie algebras with nilradical [Formula: see text] of maximal codimension. Moreover, all one-dimensional extensions of solvable Lie algebras with nilradical [Formula: see text] whose codimension is equal to one are found and we compared these solvable algebras with the solvable algebras with nilradicals that are one-dimensional central extension of algebra [Formula: see text].

The center and invariants of standard filiform Lie algebras

This paper describes the centers of the universal enveloping algebras and the invariant rings of the standard filiform Lie algebras over fields of characteristic zero and also over large enough prime characteristic. We determine explicit generators for the quotient fields and also a compact form for the generators for the invariants rings. We prove several combinatorial results concerning the Hilbert series of these algebras.

Degenerations to filiform Lie algebras of dimension 9

For complex 9-dimensional filiform Lie algebras we find via linear deformation another solvable algebra that degenerates to the given filiform algebra. This provides more evidence on a conjecture stated by Grunewald and O’Halloran.

Computation of positively graded filiform nilpotent Lie algebras in low dimensions

We describe a procedure for classifying positively graded filiform nilpotent Lie algebras in low dimensions, over R, C, and Q. Using this procedure, we find all filiform nilpotent Lie algebras of dimension 15 and lower that admit a positive grading.

Almost inner derivations of Lie algebras II

We continue the algebraic study of almost inner derivations of Lie algebras over a field of characteristic zero and determine these derivations for free nilpotent Lie algebras, for almost abelian Lie algebras, for Lie algebras whose solvable radical is abelian and for several classes of filiform nilpotent Lie algebras. We find a family of [Formula: see text]-dimensional characteristically nilpotent filiform Lie algebras [Formula: see text], for all [Formula: see text], all of whose derivations are almost inner. Finally, we compare the almost inner derivations of Lie algebras considered over two different fields [Formula: see text] for a finite-dimensional field extension.

Almost Inner Derivations of Some Nilpotent Leibniz Algebras

We investigate almost inner derivations of some finite-dimensional nilpotent Leibniz algebras. We show the existence of almost inner derivations of Leibniz filiform non-Lie algebras differing from inner derivations, we also show that the almost inner derivations of some filiform Leibniz algebras containing filiform Lie algebras do not coincide with inner derivations

Filiform Lie Algebras with Low Derived Length

We construct, for any integer n greater than or equal to 5, a family of complex filiform Lie algebras with derived length at most 3 and dimension n. We also give examples of n-dimensional filiform Lie algebras with derived length greater than 3.

Some properties of a filiform Lie algebra

Some properties of a filiform Lie algebra

Representasi Unitar Tak Tereduksi Grup Lie Dari Aljabar Lie Filiform Real Berdimensi 5

In this paper, we study a harmonic analysis of a Lie group of a real filiform Lie algebra of dimension 5. Particularly, we study its irreducible unitary representation (IUR) and contruct this IUR corresponds to its coadjoint orbits through coadjoint actions of its group to its dual space. Using induced representation of a 1-dimensional representation of its subgroup we obtain its IUR of its Lie group

On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra

In this paper, we study irreducible unitary representations of a real standard filiform Lie group with dimension equals 4 with respect to its basis. To find this representations we apply the orbit method introduced by Kirillov. The corresponding orbit of this representation is genereric orbits of dimension 2. Furthermore, we show that obtained representation of this group is square-integrable. Moreover, in such case , we shall consider its Duflo-Moore operator as multiple of scalar identity operator. In our case that scalar is equal to one.

Quasi-filiform Lie algebras of maximum length revisited

We revisit the problem of classification of quasi-filiform Lie algebras of maximum length. The lists found in the literature are either incomplete or non-explicit. En route we touch upon the classification of graded filiform Lie algebras generated by two elements of degree one and three, respectively, and, as a byproduct, obtain a classification of N-graded infinite-dimensional Lie algebras with one-dimensional homogeneous components of coclass two.

A distinguished example of filiform deformation

We exhibit the first example of a non-rigid complex filiform Lie algebra with all its codimension 1 ideals being characteristically nilpotent by constructing an explicit nontrivial filiform deformation of it.

Minimal representations of filiform Lie algebras and their application for construction of Leibniz algebras

In this paper we find minimal faithful representations of several classes of filiform Lie algebras by means of strictly upper-triangular matrices. We investigate Leibniz algebras whose corresponding Lie algebras L∕I are filiform Lie algebras such that the action I×L∕I→I gives rise to a minimal faithful representation of a filiform Lie algebra, where I is the ideal generated by the squares of the elements of the algebra L. The classification up to isomorphism of such Leibniz algebras is given for low-dimensional cases.

The Maximal Abelian Dimension of a Lie Algebra, Rentschler’s Property and Milovanov’s Conjecture

A finite dimensional Lie algebra L with magic number c(L) is said to satisfy Rentschler’s property if it admits an abelian Lie subalgebra H of dimension at least c(L) − 1. We study the occurrence of this new property in various Lie algebras, such as nonsolvable, solvable, nilpotent, metabelian and filiform Lie algebras. Under some mild condition H gives rise to a complete Poisson commutative subalgebra of the symmetric algebra S(L). Using this, we show that Milovanov’s conjecture holds for the filiform Lie algebras of type Ln, Qn, Rn, Wn and also for all filiform Lie algebras of dimension at most eight. For the latter the Poisson center of these Lie algebras is determined.