ON SIMPLE FILIPPOV SUPERALGEBRAS OF TYPE A(n, n)

Beites, Patrícia Damas; Pozhidaev, A. P.

doi:10.1142/s1793557108000394

Cited by 9 publications

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“…A description of simple Filippov superalgebras of type B(m, n) was already obtained in [10], [12] and [11]. The same problem concerning Filippov superalgebras of type A(m, n) with m = n has recently been solved in [1]. The present work represents one more step towards the classification of finite-dimensional simple Filippov superalgebras of type A(m, n) over an algebraically closed field of characteristic zero.…”

Section: Introductionmentioning

confidence: 61%

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On simple Filippov superalgebras of type A(0,n)

Beites¹,

Pozhidaev²

2010

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Section: Introductionmentioning

confidence: 61%

“…Let L = L0 ⊕ L1 be an n-ary anticommutative superalgebra. A subsuperalgebra (1) = 0 and L lacks ideals other than 0 or L.…”

Section: Introductionmentioning

confidence: 99%

On simple Filippov superalgebras of type A(0,n)

Beites¹,

Pozhidaev²

2010

Preprint

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“…Some examples of n-ary color algebras are Lie and Jordan superalgebras [25,26], Lie color algebras [30], Leibniz color algebras [23], Filippov (n-Lie) superalgebras [7,21,[33][34][35]] and 3-Lie color algebras [36]. Let us give definitions of some color algebras.…”

Section: 2mentioning

confidence: 99%

n-Ary Generalized Lie-Type Color Algebras Admitting a Quasi-multiplicative Basis

Barreiro

Martín

Kaygorodov

et al. 2018

Algebr Represent Theor

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The class of generalized Lie-type color algebras contains the ones of generalized Lie-type algebras, of n-Lie algebras and superalgebras, commutative Leibniz n-ary algebras and superalgebras, among others. We focus on the class of generalized Lie-type color algebras L admitting a quasi-multiplicative basis, with restrictions neither on the dimensions nor on the base field F and study its structure. If we write L = V⊕W with V and 0 = W linear subspaces, we say that a basis of homogeneous elements B = {e i } i∈I of W is quasimultiplicative if given 0 < k < n, for i 1 , . . . , i k ∈ I and σ ∈ S n satisfies e i1 , . . . , e i k , V, . . . , V σ ⊂ Fe jσ for some j σ ∈ I; the product of elements of the basis e i1 , . . . , e in belongs to Fe j for some j ∈ I or to V, and a similar condition is verified for the product V, . . . , V . We state that if L admits a quasi-multiplicative basis then it decomposes as L = U ⊕ ( J k ) with any J k a well described color gLt-ideal of L admitting also a quasi-multiplicative basis, and U a linear subspace of V. Also the minimality of L is characterized in terms of the connections and it is shown that the above direct sum is by means of the family of its minimal color gLt-ideals, admitting each one a µ-quasi-multiplicative basis inherited by the one of L.F e i 1 , . . . , e in .Corollary 24. Suppose L is centerless and V is tight, then L decomposes as the direct sum of the color gLt-ideals given in Proposition 17,Proof. By Theorem 21, since U = 0, we just have to show the direct character of the sum. Givenusing the fact J [i] , J [h] , L, . . . , L σ = 0 for [i] = [h] and any σ ∈ S n we obtain x, J [i] , L, . . . , L σ = x, [j] ∈ I/ ∼ j ≁ i J [h] , L, . . . , L σ = 0.

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“…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 1 The authors were supported by RFBR 15-51-04099. The first author was supported by FAPESP 14/24519-8. condition QDer = End.…”

Section: §0 Introductionmentioning

confidence: 99%

“…The main examples of color n-ary algebras are color Lie algebras [3,25], color Leibniz algebras [6], Filippov (n-Lie) superalgebras [1,2,27,29,30] and 3-Lie colour algebras [35]. Let T = g∈G T g be a color algebra.…”

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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ON SIMPLE FILIPPOV SUPERALGEBRAS OF TYPE A(n, n)

Abstract: It is proved that there exist no simple finite-dimensional Filippov superalgebras of type A(n, n) over an algebraically closed field of characteristic 0.

Cited by 9 publications

References 5 publications

On simple Filippov superalgebras of type A(0,n)

On simple Filippov superalgebras of type A(0,n)

n-Ary Generalized Lie-Type Color Algebras Admitting a Quasi-multiplicative Basis

Generalized derivations of (color)n-ary algebras

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