On Derivations of the Ternary Malcev Algebra <i>M</i> <sub>8</sub>

Pojidaev, Alexandre P.; Saraiva, Paulo

doi:10.1080/00927870600850743

Cited by 15 publications

(19 citation statements)

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“…It is easy to see (see, for example, [25]) that in Cayley-Dikson algebra C, a basis can be chosen in such way that the multiplication of the basic elements of the algebra will be represented by the following table: e 2 = e 1 e 2 = e 5 e 6 = e 7 e 8 = e 3 e 4 , e 3 = e 1 e 3 = e 7 e 6 = e 4 e 2 = e 8 e 5 , e 4 = e 1 e 4 = e 2 e 3 = e 6 e 8 = e 7 e 5 , e 5 = e 1 e 5 = e 6 e 2 = e 4 e 7 = e 3 e 8 , e 6 = e 1 e 6 = e 2 e 5 = e 8 e 4 = e 3 e 7 , e 7 = e 1 e 7 = e 5 e 4 = e 8 e 2 = e 6 e 3 , e 8 = e 1 e 8 = e 2 e 7 = e 4 e 6 = e 5 e 3 .…”

Section: Main Lemmasunclassified

Δ-Derivations OF SEMISIMPLE FINITE-DIMENSIONAL STRUCTURABLE ALGEBRAS

Kaygorodov

Okhapkina

2014

J. Algebra Appl.

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Section: Main Lemmasunclassified

Δ-Derivations OF SEMISIMPLE FINITE-DIMENSIONAL STRUCTURABLE ALGEBRAS

Kaygorodov

Okhapkina

2014

J. Algebra Appl.

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“…The class of n-ary Mal tsev algebras was defined in [24] as a natural class of n-ary algebras that contains the class of vector product n-ary algebras. The derivations of the ternary algebra M 8 were described in [29], and in [30] its root decomposition was constructed and the structure of a Z 3 -gradation was introduced. At the present time, the only known example of a simple n-ary Mal tsev algebra that is not a Filippov algebra is the simple ternary Mal tsev algebra M 8 , which arises on the 8-dimensional composition algebra.…”

Section: Corollary 3 Over An Algebraically Closed Field Of Charactermentioning

confidence: 99%

“…Then A becomes a ternary Mal tsev algebra [24] In [29], the derivations of the ternary Mal tsev algebra M 8 were described. It was shown therein that every derivation is inner, i.e.,…”

Section: Corollary 3 Over An Algebraically Closed Field Of Charactermentioning

confidence: 99%

“…License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use To complete the proof of the theorem, it remains to recall that the structure of the derivation algebra of the ternary Mal tsev algebra M 8 was found in [29], namely, isomorphism with the algebra B 3 was proved. Thus, it is easily seen that Δ(M 8 ) ∼ = B 3 .…”

Section: Corollary 3 Over An Algebraically Closed Field Of Charactermentioning

confidence: 99%

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On $(n+1)$-ary derivations of simple $n$-ary Mal′tsev algebras

Кайгородов¹

2014

St. Petersburg Math. J.

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The (n + 1)-ary derivations of simple non-Lie n-ary Mal tsev algebras are described in the case of the binary algebra M 7 and the ternary algebra M 8 . As a consequence, a description is obtained for the 3-ary derivations of simple non-Lie Mal tsev algebras and simple finite-dimensional non-Lie binary-Lie algebras. Examples of semisimple Mal tsev algebras having nontrivial 3-ary derivations are given. §1. IntroductionOne way of generalizing the derivations is a δ-derivation. By a δ-derivation of an algebra A, where δ is a fixed element of the ground field, we mean a linear mappingfor arbitrary x, y ∈ A. In one's time, δ-derivations have been studied in the papers [1,2,3,4,5,6,7,8,9,10,11,12], which were devoted to the description of δ-derivations of prime Lie [1, 2], prime alternative and Mal tsev [3] algebras, simple [5, 6] and prime [7] Lie superalgebras, semisimple finite-dimensional Jordan algebras [4, 6] and superalgebras [4, 6, 8, 9], Filippov algebras of small dimensions and simple finite-dimensional Filippov algebras [12], as well as the simple ternary Mal tsev algebra M 8 [12]. In particular, examples of nontrivial δ-derivations were constructed for some Lie algebras [2, 7], simple Jordan superalgebras [8,9], and n-ary Filippov algebras of dimension n + 1 (see [12]). At the same time, a δ-derivation is a special case of a quasiderivation and a generalized derivation. By a generalized derivation D we mean a linear mapping such that there exist linear mappings E and F with D(xy) = E(x)y + xF (y)for arbitrary x, y ∈ A. If, moreover, E = F , then D is a quasiderivation. The triples (D, E, F ), where D is a generalized derivation and E and F are linear mapping associated with it are called ternary derivations. Note that, in the general case, E and F may fail to be generalized derivations. But in many interesting cases they are, and in [13] the simple unital algebras for which E and F are generalized derivations were described under some restrictions. Quasiderivations, generalized derivations, and ternary derivations were considered in [14,15,16,17,18,19,20,21,22]. In particular, generalized and ternary derivations were studied for Lie algebras [14], Lie superalgebras [15], associative algebras [16, 17], generalized Cayley-Dickson algebras [18], and Jordan algebras [20].

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“…Filippov algebras were also known before under the names Nambu-Lie gebras and Nambu algebras. We may also remark that Filippov algebras are a particular case of n-ary Malcev algebras (see, for example, [11]). …”

Section: Introductionmentioning

confidence: 99%