On simple Filippov superalgebras of type $B(0, n)$, II

Pojidaev, Alexander P.; Saraiva, Paulo

doi:10.4171/pm/1835

Cited by 5 publications

(2 citation statements)

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“…Otherwise, if α 1 = −δ k and α 2 = −δ j , then δ k (u) 1 and δ j (u) 2. For the case s = 3, if we analyze all possibilities for w in (8) and make use of just the weight argument we arrive at…”

Section: Main Theoremmentioning

confidence: 99%

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Simple Filippov superalgebras of type B(m, n)

Pozhidaev

2008

Algebra Logic

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Section: Main Theoremmentioning

confidence: 99%

“…Consequently, there exist no simple Filippov superalgebras of type B(m, n) over an algebraically closed field of characteristic 0. Throughout the third section, it is assumed that m = 0 (the case with m = 0 was previously treated in [7,8]). …”

Section: Introductionmentioning

confidence: 99%

Simple Filippov superalgebras of type B(m, n)

Pozhidaev

2008

Algebra Logic

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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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