ON SIMPLE FILIPPOV SUPERALGEBRAS OF TYPE B(0,n)

Pojidaev, Alexandre P.

doi:10.1142/s0219498803000520

Cited by 10 publications

(22 citation statements)

References 4 publications

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“…If k = 2 then a = 1 and [10], we may assume that 1 ⊗ v is odd. Since the action of g δ on g −δ ⊗ v provides a nonzero element and g −δ ⊗ v is even, u i = g −δ ⊗ v, f or i = 1, .…”

Section: Lie Superalgebra B(0 N)mentioning

confidence: 99%

“…The case of odd generator requires techniques different from one that was used in the even case. In the present work we eliminate the assumption for the generator to be even, and prove a theorem (analogous to the main theorem of [10]) for the general case.…”

Section: Introductionmentioning

confidence: 98%

“…POJIDAEV AND P. SARAIVA facts, in this article we consider the n-ary Filippov superalgebras with n ≥ 3 and with nonzero even and odd parts. In [10], it was proved that there are no simple finite-dimensional Filippov superalgebras with multiplication Lie superalgebra isomorphic to B(0, n) under assumption that a generator of a module over B(0, n) is even. The case of odd generator requires techniques different from one that was used in the even case.…”

Section: Introductionmentioning

confidence: 99%

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

Pojidaev¹,

Saraiva²

2009

Port. Math.

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Section: Lie Superalgebra B(0 N)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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

Pojidaev¹,

Saraiva²

2009

Port. Math.

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“…. , x n−1 ), x i ∈ F. L(F) is called the multiplication algebra of an algebra F. LEMMA 1.1 [7]. Let F = F0 ⊕ F1 be a simple finite-dimensional Filippov superalgebra over a field of characteristic 0 with F1 = 0.…”

Section: Reduction To Lie Superalgebrasmentioning

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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ON SIMPLE FILIPPOV SUPERALGEBRAS OF TYPE B(0,n)

Abstract: We show that there exist no simple finite-dimensional Filippov superalgebras of type B(0,n) over an algebraically closed field of characteristic 0.

Cited by 10 publications

References 4 publications

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

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

Simple Filippov superalgebras of type B(m, n)

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

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