Generalized Derivations of Multiplicative n-Ary Hom-
                
                  
                
                $$\Omega $$
                
                  
                    Ω
                  
                
               Color Algebras

Beites, Patrícia Damas; Kaygorodov, Ivan; Popov, Yury

doi:10.1007/s40840-017-0486-8

Cited by 13 publications

(7 citation statements)

References 44 publications

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“…We note here that given a variety of (n-ary) algebras Ω defined by a family of identities {f i } i∈I we may obtain the identities {(f i ) Hom co } i∈I of the corresponding variety of color Hom-Ω algebras by twisting the defining identities by homomorphisms, usually called the twisting maps. The details of this process can be found in the papers [9,27].…”

Section: 3mentioning

confidence: 99%

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Split regular Hom-Leibniz color 3-algebras

Kaygorodov

Popov

2019

Colloq. Math.

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We introduce and describe the class of split regular Hom-Leibniz color 3-algebras as the natural extension of the class of split Lie algebras, split Lie superalgebras, split Lie color algebras, split regular Hom-Lie algebras, split regular Hom-Lie superalgebras, split regular Hom-Lie color algebras, split Leibniz algebras, split Leibniz superalgebras, split Leibniz color algebras, split regular Hom-Leibniz algebras, split regular Hom-Leibniz superalgebras, split regular Hom-Leibniz color algebras, split Lie 3-algebras, split Lie 3-superalgebras, split Lie color 3-algebras, split regular Hom-Lie 3-algebras, split regular Hom-Lie 3superalgebras, split regular Hom-Lie color 3-algebras, split Lie triple systems, split Lie triple supersystems, split Lie triple color systems, split regular Hom-Lie triple systems, split regular Hom-Lie triple supersystems, split regular Hom-Lie triple color systems, split Leibniz 3-algebras, split Leibniz 3-superalgebras, split Leibniz color 3-algebras, split regular Hom-Leibniz 3-algebras, split regular Hom-Leibniz 3-superalgebras, and some other algebras. More precisely, we show that any of such split regular Hom-Leibniz color 3-algebras T is of the form T = U + j I j , with U a subspace of the 0-root space T 0 , and I j an ideal of T satisfying for j = k : [T, I j , I k ] + [I j , T, I k ] + [I j , I k , T ] = 0.Moreover, if T is of maximal length, we characterize the simplicity of T in terms of a connectivity property in its set of non-zero roots.

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Section: 3mentioning

confidence: 99%

“…In the paper [42] Yau gave a general construction method of Hom-type algebras starting from usual algebras and a twisting self-map. For information on various types of Hom-algebras see [5,8,9,33,35,39].…”

Section: Introductionmentioning

confidence: 99%

Split regular Hom-Leibniz color 3-algebras

Kaygorodov

Popov

2019

Colloq. Math.

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“…Let ǫ be a bicharacter of G. Given two homogeneous elements x, y ∈ L we set ǫ(x, y) := ǫ(deg(x), deg(y)). Now we recall the notion of color n-ary Ω-algebra for an arbitrary family of polynomial identities Ω (see [6,27] for more details).…”

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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“…Generalized derivations of n-BiHom-Lie algebras were studied in [89]. Generalized derivations of multiplicative n-ary Hom-Ω color algebras were studied in [90]. Cohomology of Hom-Leibniz and n-ary Hom-Nambu-Lie superalgebras was considered in [91] Generalized derivations and Rota-Baxter operators of n-ary Hom-Nambu superalgebras were considered in [92].…”

Section: Introductionmentioning

confidence: 99%

On Properties and Classification of a Class of 4-Dimensional 3-Hom-Lie Algebras with a Nilpotent Twisting Map

Kitouni,

Silvestrov

2024

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The aim of this work is to investigate the properties and classification of an interesting class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map α and eight structure constants as parameters. Derived series and central descending series are studied for all algebras in this class and are used to divide it into five non-isomorphic subclasses. The levels of solvability and nilpotency of the 3-Hom-Lie algebras in these five classes are obtained. Building upon that, all algebras of this class are classified up to Hom-algebra isomorphism. Necessary and sufficient conditions for multiplicativity of general (n+1)-dimensional n-Hom-Lie algebras, as well as for algebras in the considered class, are obtained in terms of the structure constants and the twisting map. Furthermore, for some algebras in this class, it is determined whether the terms of the derived and central descending series are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals.

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Generalized Derivations of Multiplicative n-Ary Hom- $$\Omega $$ Ω Color Algebras

Cited by 13 publications

References 44 publications

Split regular Hom-Leibniz color 3-algebras

Split regular Hom-Leibniz color 3-algebras

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

On Properties and Classification of a Class of 4-Dimensional 3-Hom-Lie Algebras with a Nilpotent Twisting Map

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