On Identities in Hom-Malcev Algebras

Issa, A. Nourou

doi:10.24330/ieja.266209

Cited by 4 publications

(6 citation statements)

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“…We assume in this paper that all Hom-algebras are multiplicative. In [7] an identity, equivalent to the Hom-Malcev identity…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…In [7] it is proved that, in an anticommutative Hom-algebra (A, [, ], α), the Hom-Malcev identity (2) is equivalent to the identity…”

Section: Proofsmentioning

confidence: 99%

“…Combining Theorem 1.1 and Proposition 2.1, we get the following Corollary 2.2. In an anticommutative Hom-algebra (A, * , α), the Hom-Malcev identity (2) is equivalent to (7), with "{, , }" defined by (5).…”

Section: Proofsmentioning

confidence: 99%

“…2 (u), α 2 (v), {x, y, w}}= {x, y, u}α 2 (v) * α 3 (w)−α 2 (v)α 2 (w) * α({x, y, u})−α 2 (w){x, y, u} * α 3 (v) +α 2 (u){x, y, v} * α 3 (w)−{x, y, v}α 2 (w) * α 3 (u)−α 2 (w)α 2 (u) * α({x, y, v}) +α 2 (u)α 2 (v) * α({x, y, w})−α 2 (v){x, y, w} * α 3 (u)−{x, y, w}α 2 (u) * α 3 (v) (by (3)) = ({x, y, u} * α 2 (v) + α 2 (u) * {x, y, v}) * α 3 (w) + α 2 (u)α 2 (v) * α({x, y, w}) −({x, y, v} * α 2 (w) + α 2 (v) * {x, y, w}) * α 3 (u) − α 2 (v)α 2 (w) * α({x, y, u}) −({x, y, w} * α 2 (u) + α 2 (w) * {x, y, u}) * α 3 (v) − α 2 (w)α 2 (u) * α({x, y, v}) = {α(x), α(y), u * v} * α 3 (w) + α 2 (u * v) * {α(x), α(y), α(w)} −{α(x), α(y), v * w} * α 3 (u) − α 2 (v * w) * {α(x), α(y), α(u)} −{α(x), α(y), w * u} * α 3 (v) − α 2 (w * u) * {α(x), α(y), α(v)} (by(7) and multiplicativity) = {α 2 (x), α 2 (y), uv * α(w)} − {α 2 (x), α 2 (y), vw * α(u)} −{α 2 (x), α 2 (y), wu * α(v)} (by (7)) = {α 2 (x), α 2 (y), {u, v, w}} (by (3)) and thus we get (HLY8).…”

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A note on relations between Hom-Malcev algebras and Hom-Lie-Yamaguti algebras

Gaparayi¹,

Issa²

2015

Preprint

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“…We assume in this paper that all Hom-algebras are multiplicative. In [7] an identity, equivalent to the Hom-Malcev identity…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…In [7] it is proved that, in an anticommutative Hom-algebra (A, [, ], α), the Hom-Malcev identity (2) is equivalent to the identity…”

Section: Proofsmentioning

confidence: 99%

Section: Proofsmentioning

confidence: 99%

mentioning

confidence: 99%

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A note on relations between Hom-Malcev algebras and Hom-Lie-Yamaguti algebras

Gaparayi¹,

Issa²

2015

Preprint

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“…is equivalent to (2.2) in any anticommutative Hom-algebra (A, * , α) ([32], Proposition 2.7). In [10], it is proved that in any anticommutative Hom-algebra (A, * , α), the Hom-Malcev identity (2.2) is equivalent to…”

Section: Some Basics On Hom-algebrasmentioning

confidence: 99%

Hom-Lie Triple System and Hom-Bol Algebra Structures on Hom-Maltsev and Right Hom-Alternative Algebras

Attan

Issa

2018

International Journal of Mathematics and Mathematical Sciences

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Every multiplicative Hom-Malcev algebra has a natural multiplicative Hom-Lie triple system structure. Moreover, there is a natural Hom-Bol algebra structure on every multiplicative Hom-Malcev algebra and on every multiplicative right (or left) Homalternative algebra. 1 IntroductionThe study of Lie triple systems (Lts) on their own as algebraic objects started from Jacobson's work [11] and developed further by, e.g., Lister [13], Yamaguti [29] and other mathematicians. The interplay between Lts and the differential geometry of symmetric spaces is now folk (see, e.g., [12], [15]). Lts constitute examples of ternary algebras. If (g, [, ]) is a Lie algebra, then (g, [, , ]) is a Lts, where [x, y, z] := [[x, y], z] (see [11], [12], [15]). Another construction of Lts from binary algebras is the one from Malcev algebras found by Loos [14].Malcev algebras were introduced by Mal'tsev [20] in a study of commutator algebras of alternative algebras and also as a study of tangent algebras to local smooth Moufang loops. Mal'tsev used the name "Moufang-Lie algebras" for these nonassociative algebras while Sagle [25] introduced the term "Malcev algebras". Equivalent defining identities of Malcev algebras are pointed out in [25].Alternative algebras, Malcev algebras and Lts (among other algebras) received a twisted generalization in the development of the theory of Hom-algebras during these latest years. The forerunner of the theory of Hom-algebras is the Hom-Lie algebra introduced by Hartwig, Larsson and Silvestrov in [7] in order to describe the structure of some deformation of the Witt algebra and the Virasoro algebra. It is well-known that Lie algebras are related to associative algebras via the commutator bracket construction. In the search of a similar construction for Hom-Lie algebras, the notion of a Hom-associative algebra is introduced by Makhlouf and Silvestrov in [18], where it is proved that a Hom-associative algebra gives rise to a Hom-Lie algebra via the commutator bracket construction. Since then, various Hom-type structures are considered (see, e.g., [1], [2], [4]-[6], [9], [16]-[19], [32]-[35]). Roughly speaking, Hom-algebraic structures are corresponding ordinary algebraic structures whose defining identities are twisted by a linear self-map. A general method for constructing a Hom-type algebra from the ordinary type of algebra with a linear self-map is given by Yau in [31].In [1], [33], n-ary Hom-algebra structures generalizing n-ary algebras of Lie type or associative type were considered. In particular, generalizations of n-ary Nambu or Nambu-Lie algebras, called n-ary Hom-Nambu and Hom-Nambu-Lie algebras respectively, were introduced in [1] while Hom-Jordan were defined in [17] and Hom-Lie triple systems (Hom-Lts) were introduced in [33] (here, another definition of a Hom-Jordan algebra is given). It is shown [33] that Hom-Lts are ternary Hom-Nambu algebras with additional properties, that Hom-Lts arise also from Hom-Jordan triple systems or from other Hom-type algebras.

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Matched Pair and Manin Triple of Hom-Malcev Algebra

李进圆¹

2019

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On Identities in Hom-Malcev Algebras

Abstract: Abstract. In an anticommutative multiplicative Hom-algebra, an identity, equivalent to the Hom-Malcev identity, is found.Mathematics Subject Classification (2010): 17A30, 17D10

Cited by 4 publications

References 22 publications

A note on relations between Hom-Malcev algebras and Hom-Lie-Yamaguti algebras

A note on relations between Hom-Malcev algebras and Hom-Lie-Yamaguti algebras

Hom-Lie Triple System and Hom-Bol Algebra Structures on Hom-Maltsev and Right Hom-Alternative Algebras

Matched Pair and Manin Triple of Hom-Malcev Algebra

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