Associative Algebras Admitting a Quasi-multiplicative Basis

Martín, Antonio

doi:10.1007/s10468-014-9482-y

Cited by 8 publications

(3 citation statements)

References 14 publications

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“…and such a problem (not necessarily for group algebras) has been subsequently considered by several authors: see e.g. [1,2,5,6,8,10,15,18]. In particular, it is still an open problem whether a group algebra F G has an f.m.b.…”

Section: Introductionmentioning

confidence: 99%

On Filtered Multiplicative Bases of Some Associative Algebras

Bovdi¹,

Grishkov²,

Siciliano

2014

Algebr Represent Theor

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Abstract. We deal with the existing problem of filtered multiplicative bases of finite-dimensional associative algebras. For an associative algebra A over a field, we investigate when the property of having a filtered multiplicative basis is hereditated by homomorphic images or by the associated graded algebra of A. These results are then applied to some classes of group algebras and restricted enveloping algebras.

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Section: Introductionmentioning

confidence: 99%

On Filtered Multiplicative Bases of Some Associative Algebras

Bovdi¹,

Grishkov²,

Siciliano

2014

Algebr Represent Theor

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“…One may expect that there are problems that are naturally and more simply formulated by exploiting multiplicative bases. There are many works about the study of algebras with a multiplicative basis [25,36,37,42,51,52,73,111,196,208]. Calderón introduced the notion of quasi-multiplicative bases as a generalization of the notion of multiplicative bases in [42] and described the structure of associative algebras with a quasi-multiplicative basis.…”

Section: N-ary Algebras With a Multiplicative Type Basismentioning

confidence: 99%

“…The rest of our results in the present topic are dedicated to a generalization of the paper [42] of Calderón. Namely, in a joint work together with Barreiro, Calderón, and Sánchez [22], we obtained an analog of Calderón´s results in the class of color generalized Lie type n-ary algebras.…”

Section: N-ary Algebras With a Multiplicative Type Basismentioning

confidence: 99%

Non-associative algebraic structures: classification and structure

Kaygorodov

2023

Communications in Mathematics

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

Barreiro

Martín

Kaygorodov

et al. 2018

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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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Associative Algebras Admitting a Quasi-multiplicative Basis

Cited by 8 publications

References 14 publications

On Filtered Multiplicative Bases of Some Associative Algebras

On Filtered Multiplicative Bases of Some Associative Algebras

Non-associative algebraic structures: classification and structure

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

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