Ternary derivations of finite-dimensional real division algebras

Gestal, Clara Jiménez; Pérez-Izquierdo, José M.

doi:10.1016/j.laa.2007.11.019

Cited by 27 publications

(14 citation statements)

References 22 publications

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“…The concept of a ternary derivation goes back to the so-called triality principle, which states that every element of the orthogonal Lie algebra of skew-symmetric operators of the Cayley-Dickson algebra under the composition form is the principal component of a ternary derivation. In [20,31] ternary derivations have been used to study some unital associative algebras; recently, Shestakov [35] has described the ternary derivations of separable associative and Jordan algebras.…”

Section: Ternary Derivationsmentioning

confidence: 99%

“…We say that a dialgebra is semiprime if it does not contain di-ideals which left and right square products are zero; that is, I I = I I = 0 implies I = 0 for every di-ideal I . (a x, y, z) ≡ a (x, y, z) , (20) (a x, y, z) ≡ a (x, y, z) ,…”

Section: ((Ab)c)d ((Ab)d)c ((Ac)b)d ((Ac)d)b ((Ad)b)c ((Ad)c)b mentioning

confidence: 99%

“…Since (x, y, z) ∈ N(D), an application of (20) gives p ≡ ((x, y, z) t, u, v) . On the other hand from (T1) we obtain…”

Section: ((Ab)c)d ((Ab)d)c ((Ac)b)d ((Ac)d)b ((Ad)b)c ((Ad)c)b mentioning

confidence: 99%

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On the definitions of nucleus for dialgebras

Sánchez-Ortega

2013

Journal of Algebra

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Communicated by Alberto ElduqueIn memory of Jean-Louis Loday MSC: primary 17D10 secondary 17A20, 17A30, 17A32, 17D05 Keywords:Nonassociative algebras and dialgebras Nucleus Ternary derivations Computer algebra Malcev dialgebras were introduced recently by Bremner, Peresi and Sánchez-Ortega. In the present paper, we continue their study by introducing the notion of the generalized alternative dinucleus of a 0-dialgebra. A general conjecture about the speciality of Malcev dialgebras in terms of this dinucleus is formulated. In the last section, we introduce the appropriate generalization of the associative nucleus for dialgebras, and prove an analogue of Kleinfeld's theorem for the setting of dialgebras.

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Section: Ternary Derivationsmentioning

confidence: 99%

Section: ((Ab)c)d ((Ab)d)c ((Ac)b)d ((Ac)d)b ((Ad)b)c ((Ad)c)b mentioning

confidence: 99%

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On the definitions of nucleus for dialgebras

Sánchez-Ortega

2013

Journal of Algebra

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“…More recently, Jiménez-Gestal and Pérez-Iquierdo [15] have studied the relationship that exists between the isotopisms of a finite-dimensional real division algebra and the Lie algebra of its ternary derivations. On the other hand, Allison et al [2,3] have recently studied isotopes of a class of graded Lie algebras called Lie tori, but the notion of isotopism that they use is quite different from the conventional one.…”

Section: Introductionmentioning

confidence: 99%

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

2016

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Since the introduction of the concept of isotopism of algebras by Albert in 1942, a prolific literature on the subject has been developed for distinct types of algebras. Nevertheless, there barely exists any result on the problem of distributing Lie algebras into isotopism classes. The current paper is a first step to deal with such a problem. Specifically, we define a new series of isotopism invariants and we determine explicitly the distribution into isotopism classes of n-dimensional filiform Lie algebras, for n ≤ 7. We also deal with the distribution of such algebras into isomorphism classes, for which we confirm some known results and we prove that there exist p + 8 isomorphism classes of seven-dimensional filiform Lie algebras over the finite field F p if p = 2.

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“…Generalized derivations of simple algebras and superalgebras were investigated in [11,23,31,32]. Perez-Izquierdo and Jimenez-Gestal used the generalized derivations to study non-associative algebras [12,26]. Derivations and generalized derivations of n-ary algebras were considered in [2, 18-21, 28, 33] and other.…”

Section: §0 Introductionmentioning

confidence: 99%

Generalized derivations of (color)n-ary algebras

Kaygorodov

Popov

2015

Linear and Multilinear Algebra

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Abstract. We generalize the results of Leger and Luks about generalized derivations of Lie algebras to the case of color n-ary Ω-algebras. Particularly, we prove some properties of generalized derivations of color n-ary algebras; prove that a quasiderivation algebra of a color n-ary Ω-algebra can be embedded into the derivation algebra of a larger color n-ary Ω-algebra, and describe (anti)commutative n-ary algebras satisfying the condition QDer = End. §0 Introduction It is well known that the algebras of derivations and generalized derivations are very important in the study of Lie algebras and its generalizations. There are many generalizations of derivations (for example, Leibniz derivations [22] and Jordan derivations [13]). The notion of δ-derivations appeared in the paper of Filippov [8], he studied δ-derivations of prime Lie and Malcev algebras [9,10]. After that, δ-derivations of Jordan and Lie superalgebras were studied in [14][15][16][17]36] and many other works. The notion of generalized derivation is a generalization of δ-derivation. The most important and systematic research on the generalized derivations algebras of a Lie algebra and their subalgebras was due to Leger and Luks [24]. In their article, they studied properties of generalized derivation algebras and their subalgebras, for example, the quasiderivation algebras. They have determined the structure of algebras of quasiderivations and generalized derivations and proved that the quasiderivation algebra of a Lie algebra can be embedded into the derivation algebra of a larger Lie algebra. Their results were generalized by many authors. For example, Zhang and Zhang [34] generalized the above results to the case of Lie superalgebras; Chen, Ma, Ni and Zhou considered the generalized derivations of color Lie algebras, Hom-Lie superalgebras and Lie triple systems [3][4][5]. Generalized derivations of simple algebras and superalgebras were investigated in [11,23,31,32]. Perez-Izquierdo and Jimenez-Gestal used the generalized derivations to study non-associative algebras [12,26]. Derivations and generalized derivations of n-ary algebras were considered in [2, 18-21, 28, 33] and other. For example, Williams proved that, unlike the case of binary algebras, for any n ≥ 3 there exist a non-nilpotent n-Lie algebra with invertible derivation [33], Kaygorodov described (n + 1)-ary derivations of simple n-ary Malcev algebras [20] and generalized derivations algebra of semisimple Filippov algebras over an algebraically closed field of characteristic zero [21].The main purpose of our work is to generalize the results of Leger and Luks to the case of color n-ary algebras. Particularly, we prove some properties of generalized derivations of color n-ary algebras; prove that the quasiderivation algebra of a color n-ary Ω-algebra can be embedded into the derivation algebra of a large color n-ary Ω-algebra. We describe all nonabelian n-ary (anti)commutative algebras with the

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Ternary derivations of finite-dimensional real division algebras

Cited by 27 publications

References 22 publications

On the definitions of nucleus for dialgebras

On the definitions of nucleus for dialgebras

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

Generalized derivations of (color)n-ary algebras

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