Isomorphism and Isotopism Classes of Filiform Lie Algebras of Dimension up to Seven Over Finite Fields

Falcón, Óscar J.; Ganfornina, Raúl Manuel Falcón; Núñez, Juan

doi:10.1007/s00025-016-0616-x

Cited by 4 publications

(3 citation statements)

References 19 publications

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“…In this case, a consequence of Engel Theorem for filiform Lie algebras of dimension 7 over a field of characteristic 2 assures the existence of two basis B 1 and B 2 where, besides of the bracket products of the adapted basis, there exist these other ones [6] As a consequence, the following classification is obtained Theorem 6. Up to isomorphism, there exist fifteen 7-dimensional filiform Lie algebras over Z/2Z.…”

Section: -Dimensional Filiform Lie Algebras Over Z/2zmentioning

confidence: 97%

Classification of Filiform Lie Algebras up to dimension 7 Over Finite Fields

Falcón¹,

Ganfornina²,

Núñez³

et al. 2016

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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This paper tries to develop a recent research which consists in using Discrete Mathematics as a tool in the study of the problem of the classification of Lie algebras in general, dealing in this case with filiform Lie algebras up to dimension 7 over finite fields. The idea lies in the representation of each Lie algebra by a certain type of graphs. Then, some properties on Graph Theory make easier to classify the algebras. As main results, we find out that there exist, up to isomorphism, six, five and five 6-dimensional filiform Lie algebras and fifteen, eleven and fifteen 7-dimensional ones, respectively, over Z/pZ, for p = 2, 3, 5. In any case, the main interest of the paper is not the computations itself but both to provide new strategies to find out properties of Lie algebras and to exemplify a suitable technique to be used in classifications for larger dimensions.

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Section: -Dimensional Filiform Lie Algebras Over Z/2zmentioning

confidence: 97%

Classification of Filiform Lie Algebras up to dimension 7 Over Finite Fields

Falcón¹,

Ganfornina²,

Núñez³

et al. 2016

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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show abstract

“…Clara Jiménez-Gestal and Pérez-Iquierdo [104] studied how isotopisms of finite-dimensional real division algebra are related to the Lie algebra of its ternary derivations. More recently, the authors [105,106] dealt with the distribution of filiform Lie algebras into isotopism classes. 4. ten isotopism classes of seven-dimensional filiform Lie algebras over any algebraically closed or finite field of characteristic two.…”

Section: Lie Algebrasmentioning

confidence: 99%

A Historical Perspective of the Theory of Isotopisms

2018

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In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.

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“…Nevertheless, the classification of Lie algebras into isotopism classes is also interesting because it allows for combining non-isomorphic algebras that share some properties that are not detected by isomorphisms. The study of classifications of Lie algebras into isotopism classes was recently initiated for filiform Lie algebras in [5]. Previously, isotopisms have also been used to classify distinct algebraic and combinatoric structures such as Jordan algebras [10], alternative algebras [2], division algebras [11], alternating forms [7], quasigroups [6] and Latin squares [8].…”

mentioning

confidence: 99%