Family of p-Filiform Lie Algebras

Cabezas, J. M.; Gómez, José Manuel; Jiménez-Merchán, A.

doi:10.1007/978-94-011-5072-9_7

Cited by 15 publications

(4 citation statements)

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“…The more the dimension increases, the more and more complex is the determination of exhaustive lists of Lie algebras, so new computation methodologies are a present field of research [21][22][23]. Cabezas et al (1998) [24] study a family of Lie algebras that they call p-filiform with dimension n and Goze's invariant (n − p, 1, ..., 1) . Since filiform algebras have Goze's invariant (n − 1, 1), they are included in the p-filiform family as one-filiform Lie algebras; analogously, the quasi-filiform algebras are the two-filiform algebras [25], and the abelian algebras are the (n − 1)-filiform algebras.…”

Section: State Of the Artmentioning

confidence: 99%

Symbolic and Iterative Computation of Quasi-Filiform Nilpotent Lie Algebras of Dimension Nine

Pérez

Jiménez

2015

Symmetry

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This paper addresses the problem of computing the family of two-filiform Lie algebra laws of dimension nine using three Lie algebra properties converted into matrix form properties: Jacobi identity, nilpotence and quasi-filiform property. The interest in this family is broad, both within the academic community and the industrial engineering community, since nilpotent Lie algebras are applied in traditional mechanical dynamic problems and current scientific disciplines. The conditions of being quasi-filiform and nilpotent are applied carefully and in several stages, and appropriate changes of the basis are achieved in an iterative and interactive process of simplification. This has been implemented by means of the development of more than thirty Maple modules. The process has led from the first family formulation, with 64 parameters and 215 constraints, to a family of 16 parameters and 17 constraints. This structure theorem permits the exhaustive classification of the quasi-filiform nilpotent Lie algebras of dimension nine with current computational methodologies.

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Section: State Of the Artmentioning

confidence: 99%

Symbolic and Iterative Computation of Quasi-Filiform Nilpotent Lie Algebras of Dimension Nine

Pérez

Jiménez

2015

Symmetry

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“…Regarding the index of nilpotency, we recall that in the theory of Lie algebras a finite dimensional filiform Lie algebra over field K is a nilpotent Lie algebra L whose nil index is maximal and equal to dim(L) − 1. The notion of p-filiform Lie algebras makes sense for p ≥ 1 (see [11]), while for p = 0 it is not possible to define that, since a Lie algebra has at least two generators. In the case of Leibniz algebras the class of null-filiform Leibniz algebras and their properties was originally introduced in [5].…”

Section: Introductionmentioning

confidence: 99%

Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

Centrone

Zargeh

2021

Algebr Represent Theor

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“…In the theory of nilpotent Lie algebras powerful techniques have been generated in naturally graded Lie algebras (for example, in cohomology description and structural properties, see Goze and Khakimdjanov (1996)) which have been applied to non-graded algebras (Cabezas and Pastor, 2005;Gómez and Jiménez-Merchán, 2002). Since finding naturally graded Leibniz algebras is always possible for nilpotent algebras, these techniques are always applicable.…”

Section: Introductionmentioning

confidence: 99%