Hom-Pre-Malcev and Hom-M-Dendriform Algebras

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The notion of dendriform algebras was introduced in 1995 by J.-L. Loday [35] in his study of algebraic K-theory. Dendriform algebras are algebras with two operations, which dichotomize the notion of associative algebras. Later the notion of tridendriform algebra were introduced by Loday and Ronco in their study of polytopes and Koszul duality (see [36]). In order to determine the algebraic structures behind a pair of commuting Rota-Baxter operators (on an associative algebra), Aguiar and Loday introduced the notion of quadri-algebra in [1]. Malcev algebras were introduced in 1955 by A.I.Malcev [46], who called these ob- jects Moufang-Lie algebras because of the connection with analytic Moufang loops. A Malcev algebra is a non-associative algebra A with an anti-symmetric multiplication [−, −] that satisﬁes, for all x,y,z ∈ A, the Malcev identity:

J(x,y, [x, z]) = [J(x,y,z),x], (1.1) where J(x,y,z) = [[x,y], z] + [[z,x],y] + [[y, z],x] is the Jacobian. In particular, Lie algebras are examples of Malcev algebras. Malcev algebras play an important role in Physics and the geometry of smooth loops. Just as the tangent algebra of a Lie group is a Lie algebra, the tangent algebra of a locally analytic Moufang loop is a Malcev algebra [26–28, 46, 53, 58], see also [17, 52, 54] for discussions about connections with physics. The notion of pre-Malcev algebra as a Malcev algebraic analogue of a pre- Lie algebra was introduced in [41]. A pre-Malcev algebra is a vector space A with a bilinear multiplication ′′·′′ such that the product [x,y]= x · y − y · x endows A with the structure of a Malcev algebra, and the left multiplication operator L·(x) : y 7→ x · y deﬁne a representation of this Malcev algebra on A. In other words, the product x · y satisﬁes the following identities for all x,y,z,t ∈ A:

[y, z] · (x · t) + [[x,y], z] · t + y · ([x, z] · t) − x · (y · (z · t)) + z · (x · (y · t)). (1.2)

In [41], in order to find a dendriform algebra whose anti-commutator is a pre- Malcev algebra, M-dendriform algebras were introduced, an O-operator (specially a Rota-Baxter operator of weight zero) on a pre-Malcev algebra or two commuting Rota- Baxter operators on a Malcev algebra were shown to give a M-dendriform algebra, and also the relationships between M-dendriform algebras and Loday algebras especially quadri-algebras was established. The theory of Hom-algebras has been initiated in [18,29,30] from Hom-Lie algebras, quasi-Hom-Lie algebras and quasi-Lie algebras, motivated by quasi-deformations of Lie algebras of vector fields, in particular q-deformations of Witt and Virasoro algebras. Hom-Lie algebras and more general quasi-Hom-Lie algebras were introduced first by Hartwig, Larsson and Silvestrov in [18] where a general approach to discretization of Lie algebras of vector fields using general twisted derivations (σ-derivations) and a

2 general method for construction of deformations of Witt and Virasoro type algebras based on twisted derivations have been developed. The general quasi-Lie algebras, con- taining the quasi-Hom-Lie algebras and Hom-Lie algebras as subclasses, as well their graded color generalization, the color quasi-Lie algebras including color quasi-hom-Lie algebras, color hom-Lie algebras and their special subclasses the quasi-Hom-Lie superalgebras and hom-Lie superalgebras, have been first introduced in [18,29–32,62]. Sub- sequently, various classes of Hom-Lie admissible algebras have been considered in [47]. In particular, in [47], the Hom-associative algebras have been introduced and shown to be Hom-Lie admissible, that is leading to Hom-Lie algebras using commutator map as new product, and in this sense constituting a natural generalization of associative algebras as Lie admissible algebras leading to Lie algebras using commutator map. Fur- thermore, in [47], more general G-Hom-associative algebras including Hom-associative algebras, Hom-Vinberg algebras (Hom-left symmetric algebras), Hom-pre-Lie algebras (Hom-right symmetric algebras), and some other Hom-algebra structures, generalizing G-associative algebras, Vinberg and pre-Lie algebras respectively, have been introduced and shown to be Hom-Lie admissible, meaning that for these classes of Hom-algebras, the operation of taking commutator leads to Hom-Lie algebras as well. Also, flexible Hom-algebras have been introduced, connections to Hom-algebra generalizations of derivations and of adjoint maps have been noticed, and some low-dimensional Hom- Lie algebras have been described. In Hom-algebra structures, defining algebra identities are twisted by linear maps. Since the pioneering works [18, 29–32, 47], Hom-algebra structures have developed in a popular broad area with increasing number of publi- cations in various directions. Hom-algebra structures include their classical counter- parts and open new broad possibilities for deformations, extensions to Hom-algebra structures of representations, homology, cohomology and formal deformations, Hom- modules and hom-bimodules, Hom-Lie admissible Hom-coalgebras, Hom-coalgebras, Hom-bialgebras, Hom-Hopf algebras, L-modules, L-comodules and Hom-Lie quasi- bialgebras, n-ary generalizations of biHom-Lie algebras and biHom-associative algebras and generalized derivations, Rota-Baxter operators, Hom-dendriform color algebras, Rota-Baxter bisystems and covariant bialgebras, Rota-Baxter cosystems, coqu- asitriangular mixed bialgebras, coassociative Yang-Baxter pairs, coassociative Yang- Baxter equation and generalizations of Rota-Baxter systems and algebras, curved O- operator systems and their connections with tridendriform systems and pre-Lie algebras, BiHom-algebras, BiHom-Frobenius algebras and double constructions, infinitesi- mal biHom-bialgebras and Hom-dendriform D-bialgebras, and category theory of Hom- algebras [2,3,5–12,15,16,19–24,29,32–34,37–40,42,45,48–51,55–57,60,61,63–65,68–72]. A Hom-type generalization of Malcev algebras, called Hom-Malcev algebras, is defined in [67], where connections between Hom-alternative algebras and Hom-Malcev algebras are given. We aim in this paper to introduce and study, through Rota- Baxter operators and O-operators, the relationship between Hom-Malcev, Hom-pre-

3 Malcev algebras and Hom-M-dendriform algebras generalizing, then, Malcev algebras, pre-Malcev algebras and M-dendriform algebras. The anti-commutator of a Hom-pre- Malcev algebra is a Hom-Malcev algebra and the left multiplication operators give a representation of this Hom-Malcev algebra, which is the beauty of such a structure. Similarly, a Hom-M-dendriform algebra gives rise to a Hom-pre-Malcev algebra, and a Hom-Malcev algebra, in the same way as Hom-L-dendriform algebra, gives rise to Hom-pre-Lie algebra and Hom-Lie algebra (see [13,14,43] for more details). In Section 2, we summarize the deﬁnitions and some key constructions of Hom- alternative algebras and Hom-Malcev algebras and we introduce the notion of O- operator of Hom-Malcev algebras that generalizes the notion of Rota-Baxter operators. In Section 3, we introduce the notion of Hom-pre-Malcev algebra, provide some properties and deﬁne the notion of a bimodule of a Hom-pre-Malcev algebra. Moreover, we develop some constructions theorems. In Section 4, we introduce the notion of Hom- M-dendriform algebra and then study some of their fundamental properties in terms of the O-operators of Hom-pre-Malcev algebras. Their relationship with Hom-Malcev algebras and Hom-alternative quadri algebras are also described.

2 Preliminaries and basics

The purpose of this section is to recall some basics about Hom-alternative algebras introduced in [44,67]. Moreover, we give the definition of Hom-Malcev algebras which may be viewed as a Hom-alternative algebra via the commutator bracket (see [67]). In this paper, all vector spaces are over a field K of characteristic 0. A Hom-algebra is a triple (A, µ, α) in which A is a vector space, µ : A⊗2 −→ A is a bilinear map and α : A −→ A is a linear map (the twisting map). Hom-algebra is said to be multiplicative if α ◦ µ = µ ◦ α⊗2. Since many of the results in this paper depend on this property, we will assume the multiplicativity property for Hom-algebras as default in the paper, and thus will not write the word multiplicative each time, for simplicity of exposition. Also, when there is no ambiguity, we denote for simplicity the multiplication and composition by concatenation. Definition 2.1 ( [44]). A Hom-alternative algebra is a Hom-algebra (A, ∗, α) satisfying for all x,y,z ∈ A,

asα(x,x,y)= asα(y,x,x) = 0, (2.1)

where asα(x,y,z) = (x ∗ y) ∗ α(z) − α(x) ∗ (y ∗ z) is the Hom-associator. Deﬁnition 2.2 ( [67]). A Hom-Malcev algebra is a Hom-algebra (A, [−, −], α) such that [−, −] is anti-symmetric, and satisﬁes the Hom-Malcev identity for all x,y,z ∈ A,

2 Jα(α(x), α(y), [x, z]) = [Jα(x,y,z), α (x)], (2.2)

4 where Jα(x,y,z) = [[x,y], α(z)] + [[y, z], α(x)] + [[z,x], α(y)] is the Jacobian of x,y,z. The Hom-Malcev identity 2.2 is equivalent to

[α([x, z]), α([y,t])] = [[[x,y], α(z)], α2 (t)] + [[[y, z], α(t)], α2 (x)] (2.3) + [[[z,t], α(x)], α2 (y)] + [[[t,x], α(y)], α2 (z)].

When α = IdA, we recover the Malcev algebra (see [46]). If the map α satisﬁes α([x,y]) = [α(x), α(y)], then the Hom-Malcev algebra is said to be multiplicative. Throughout this article we will impose multiplicative property on α because many of our results depend on it, and thus we will not write the word multiplicative each time. Hom-alternative algebras related to Hom-Malcev algebras via admissibility [67].

Theorem 2.1. Any Hom-alternative algebra (A, ∗, α) is a Hom-Malcev admissible algebra. That is (A, [−, −], α) is a Hom-Malcev algebra with the bilinear bracket deﬁned by commutator [x,y]= x ∗ y − y ∗ x for all x,y ∈ A.

Let (A, [−, −], α) and (A′, [−, −]′, α′) be two Hom-Malcev algebras. A linear map f : A → A′ is said to be a morphism of Hom-Malcev algebras if for all x,y ∈ A,

[f(x) f(y)]′ = f([x,y]) and f ◦ α = α′ ◦ f

Deﬁnition 2.3. Let (A, [−, −], α) be a Hom-Malcev algebra and V be a vector space, a linear map ρ : A −→ End(V ) is a representation of (A, [−, −], α) on V with respect to β ∈ End(V ) if for any x,y,z ∈ A,

ρ(α(x))β = βρ(x), (2.4) ρ([[x,y], α(z)])β2 = ρ(α2(x))ρ(α(y))ρ(z) − ρ(α2(z))ρ(α(x))ρ(y) + ρ(α2(y))ρ([z,x])β − ρ(α([y, z]))ρ(α(x))β. (2.5)

Proposition 2.1. Let (A, [−, −], α) be a Hom-Malcev algebra, (V, β) a vector space and ρ : A −→ End(V ) a linear map. Then (V, ρ, β) is a representation of A if and only if (A ⊕ V, [−, −]ρ, α + β) is a Hom-Malcev algebra, where [−, −]ρ and α + β are deﬁned for all x,y ∈ A, a, b ∈ V by

[x + a, y + b]ρ = [x,y]+ ρ(x)a − ρ(y)b, (α + β)(x + a) = α(x)+ β(a).

This Hom-Malcev algebra is called the semi-direct product of (A, [−, −], α) and (V, β), ⋉α,β ⋉ and denoted by A ρ V or simply A V .

5 Example 2.1. Let (A, [−, −], α) be a Hom-Malcev algebra. Then adx : A −→ End(A) deﬁned by adx(y) = [x,y], ∀x,y ∈ A, is a representation of (A, [−, −], α) on A with respect to α, which is called the adjoint representation of A. Theorem 2.2. Let (A, [−, −]) be a Malcev algebra and α : A −→ A be an algebra morphism. Then the Hom-algebra Aα = (A, [−, −]α = α ◦ [−, −], α) induced by α is a Hom-Malcev algebra. Moreover, assume that (A′, [−, −]′) is another Malcev algebra, and α′ : A′ → A′ is a Malcev algebra morphism. Let f : A → A′ be a Malcev algebra ′ ′ morphism satisfying f ◦ α = α ◦ f. Then f : Aα → Aα′ is a Hom-Malcev algebra morphism. The following result gives a procedure to construct representation of Hom-Malcev algebra by a representation Malcev algebra, morphism and linear map. Proposition 2.2. Let (A, [−, −]) be a Malcev algebra, α : A → A be a morphism on A, (V, ρ) be a representation of A and β : V → V be a linear map such that βρ(x)(a)= ρ(α(x))β(a). Then (V, ρ, β) is a representation of (A, [−, −]α, α), where ρ(x)(a)= ρ(eα(x))β(a), ∀ x ∈ A, a ∈ V. Proof. For any x,y,z ∈eA and a ∈ V , by (2.4), βρ(x)(a)= β(ρ(α(x))β(a)) = ρ(α2(x))β2(a)= ρ(α(x))β(a). Combining withe (2.5), we get: e

2 2 2 ρ([[x,y]α, α(z)]α)(β (a)) − ρ(α (x))ρ(α(y))ρ(z)(a)+ ρ(α (z))ρ(α(x))ρ(y)(a) 2 e + ρ(α (y))ρ([ex, z]α)β(ea)+ ρ(eα([y, z]α))eρ(α(x))βe(a)= e 3 e e e e β ρ([[x,y], z]) − ρ(x)ρ(y)ρ(z)+ ρ(z)ρ(x)ρ(y)+ ρ(y)ρ([x, z]) + ρ([y, z])ρ(x)(a) = 0.

Then (V, ρ, β) is a representation of (A, [−, −]α, α) on V . The followinge terminology is motivated by the notion of O-operator as a generalization of Rota-Baxter operator of weight 0. Deﬁnition 2.4. A linear map T : V → A is called an O-operator associated to a representation (V, ρ, β) of a Hom-Malcev algebra (A, [−, −], α) if for all a, b ∈ V, α ◦ T = T ◦ β and [T (a), T (b)] = T ρ(T (a))b − ρ(T (b))a . (2.6) Example 2.2. A Rota-Baxter operator of weight 0 on a Hom-Malcev algebra A is just an O-operator associated to the adjoint representation (A, ad, α), that is, R satisﬁes R ◦ α = α ◦ R and [R(x), R(y)] = R([R(x),y] + [x, R(y)]), for all x,y ∈ A.

6 3 Hom-pre-Malcev algebras

In this section, we generalize the notion of pre-Malcev algebras introduced in [41] to the Hom case and study the relationships with Hom-Malcev algebras in terms of O- operators of Hom-Malcev algebras and Hom-pre-alternative algebras. Moreover we characterize the representation of Hom-pre-Malcev algebras and provide some key constructions.

3.1 Deﬁnition and basic properties Deﬁnition 3.1. A Hom-pre-Malcev algebra is a Hom-algebra (A, ·, α) satisfying, for any x,y,z,t ∈ A and [x,y]= x · y − y · x, the identity

[α(y), α(z)] · α(x · t) + [[x,y], α(z)] · α2(t) (3.1) + α2(y) · ([x, z] · α(t)) − α2(x) · (α(y) · (z · t)) + α2(z) · (α(x) · (y · t)) = 0.

The identity (3.1) is equivalent to HP M(x,y,z,t) = 0, where for all x,y,z,t ∈ A,

HP M(x,y,z,t)= α(y · z) · α(x · t) − α(z · y) · α(x · t) + ((x · y) · α(z)) · α2(t) − ((y · x) · α(z)) · α2(t) − (α(z) · (x · y)) · α2(t) + (α(z) · (y · x)) · α2(t) (3.2) + α2(y) · ((x · z) · α(t)) − α2(y) · ((z · x) · α(t)) − α2(x) · (α(y) · (z · t)) + α2(z) · (α(x) · (y · t)).

A Hom-pre-Malcev algebra is said to be a multiplicative Hom-pre-Malcev algebra if α satisﬁes α(x · y)= α(x) · α(y), for all x,y ∈ A. Hom-pre-Malcev algebras generalize Hom-pre-Lie algebras. A Hom-pre-Lie algebra is a vector space A with a bilinear product · and a linear map α satisfying the Hom- pre-Lie identity HPL(x,y,z)= asα(x,y,z) − asα(y,x,z) = 0, for all x,y,z ∈ A, where asα(x,y,z) = (x · y) · α(z) − α(x) · (y · z) is the Hom-associator. Note that

HP M(x,y,z,t)= HPL([x,y], α(z), α(t)) − HPL([y,x], α(z), α(t)) + HPL(α(x), α(y), [z,t]) + HPL(α(y), α(z), [x,t]) − [α2(z),HPL(x,y,t)] + [α2(y),HPL(x,z,t)].

So every Hom-pre-Lie algebra is a Hom-pre-Malcev algebra. When α = IdA, Hom-pre-Malcev algebra (A, ·, α) is a pre-Malcev algebra.

7 Deﬁnition 3.2. Let (A, ·, α) and (A′, ·′, α′) be two Hom-pre-Malcev algebras. A linear map f : A → A′ is called a morphism of Hom-pre-Malcev algebras if, for all x,y ∈ A,

f(x) ·′ f(y)= f(x · y), f ◦ α = α′ ◦ f.

The following theorem provides a procedure to construct Hom-pre-Malcev algebra by a pre-Malcev algebra and a morphism.

Theorem 3.1. Let A = (A, ·) be a pre-Malcev algebra and let α : A −→ A be a morphism of A. Then Aα = (A, ·α, α) is a Hom-pre-Malcev algebra with x·αy = α(x·y). Moreover, assume that A′ = (A′, ·′) is another pre-Malcev algebra and α′ : A′ → A′ is a pre-Malcev algebra morphism of A′. Let f : A → A′ be a pre-Malcev algebra morphism ′ ′ satisfying f ◦ α = α ◦ f. Then f : Aα → Aα′ is a Hom-pre-Malcev algebra morphism.

Proof. We just show that (A, ·α, α) satisﬁes the identity (3.1) while (A, ·) satisﬁes the identity (1.2). Indeed,

α([y, z] ) · α(x · t)= α3 [y, z] · (x · t) , α α α 2 3 [[x,y]α, α(z)]α ·α α (t)= α ([[x,y], z] · t), 2 3 α (y) ·α ([x, z]α ·α α(t)) = α (y · ([x, z] · t)), 2 3 α (x) ·α (α(y) ·α (z ·α t)) = α (x · (y · (z · t))), 2 3 α (z) ·α (α(x) ·α (y ·α t)) = α (z · (x · (y · t))).

The second assertion follows from ′ ′ ′ ′ f(x ·α y)= f(α(x · y)) = α (f(x · y)) = α (f(x) · f(y)) = f(x) ·α′ f(y).

Proposition 3.1. Let (A, ·, α) be a Hom-pre-Malcev algebra. The commutator given, for all x, y ∈ A, by

[x,y]= x · y − y · x, (3.3) deﬁnes a Hom-Malcev algebra structure on A.

8 Proof. We show that the commutator (3.3) satisﬁes the identity (2.3). For x,y,z,t ∈ A,

[α([x, z]), α([y,t])] − [[[x,y], α(z)], α2 (t)] − [[[y, z], α(t)], α2 (x)] − [[[z,t], α(x)], α2 (y)] − [[[t,x], α(y)], α2 (z)] = α(x · z) · α(y · t) − α(x · z) · α(t · y) − α(z · x) · α(y · t)+ α(z · x) · α(t · y) − α(y · t) · α(x · z)+ α(y · t) · α(z · x)+ α(t · y) · α(x · z) − α(t · y) · α(z · x) − ((x · y) · α(z)) · α2(t) + ((y · x) · α(z)) · α2(t) + (α(z) · (x · y)) · α2(t) − (α(z) · (y · x)) · α2(t)+ α2(t) · ((x · y) · α(z)) − α2(t) · ((y · x) · α(z)) − α2(t) · (α(z) · (x · y)) + α2(t) · (α(z) · (y · x)) − ((y · z) · α(t)) · α2(x) + ((z · y) · α(t)) · α2(x) + (α(t) · (y · z)) · α2(x) − (α(t) · (z · y)) · α2(x) + α2(x) · ((y · z) · α(t)) − α2(x) · ((z · y) · α(t)) − α2(x) · (α(t) · (y · z)) + α2(x) · (α(t) · (z · y)) − ((z · t) · α(x)) · α2(y) + ((t · z) · α(x)) · α2(y) + (α(x) · (z · t)) · α2(y) − (α(x) · (t · z)) · α2(y)+ α2(y) · ((z · t) · α(x)) − α2(y) · ((t · z) · α(x)) − α2(y) · (α(x) · (z · t)) + α2(y) · (α(x) · (t · z)) − ((t · x) · α(y)) · α2(z) + ((x · t) · α(y)) · α2(z) + (α(y) · (t · x)) · α2(z) + (α(y) · (x · t)) · α2(z)+ α2(z) · ((t · x) · α(y)) − α2(z) · ((x · t) · α(y)) − α2(z) · (α(y) · (t · x)) + α2(z) · (α(y) · (x · t)) = HP M(x,t,y,z)+ HP M(y,x,z,t)+ HP M(z,y,t,x)+ HP M(t,z,x,y) = 0.

Deﬁnition 3.3. The Hom-Malcev algebra structure in Proposition 3.1 is called the associated Hom-Malcev algebra of the Hom-pre-Malcev algebra (A, ·, α), and the Hom- pre-Malcev algebra (A, ·, α) is called a compatible Hom-pre-Malcev algebra on the Hom- Malcev algebra (A, [−, −], α).

A Hom-pre-Malcev algebra can be viewed as a Hom-Malcev algebra whose operation decomposes into two compatible pieces. Examples of Hom-pre-Malcev algebras can be constructed from Hom-Malcev algebras with O-operators. Let (A, [−, −], α) be a Hom-Malcev algebra and T : V → A be an O-operator of A associated to a module (V, ρ, β). Deﬁne the product ” · ” by

a · b = ρ(T (a))b, ∀ a, b ∈ V. (3.4)

Proposition 3.2. With the above notations (V, ·, β) is a Hom-pre-Malcev algebra, and there exists an associated Hom-Malcev algebra structure on V given by (3.3) and T is a homomorphism of Hom-Malcev algebras. Furthermore, T (V )= {T (a), a ∈ V }⊂ A is a Hom-Malcev subalgebra of (A, [−, −], α) and (T (V ), [−, −], α) is a Hom-pre-Malcev algebra structure given, for all a, b ∈ V, by T (a) · T (b) = T (a · b). Moreover, the

9 corresponding associated Hom-Malcev algebra structure on T (V ) given by (3.3) is just a Hom-Malcev subalgebra structure of (A, [−, −], α), and T is a morphism of Hom-pre- Malcev algebras.

Proof. By the identity of O-operator (2.6), T ([a, b]T ) = T (a · b − b · a) = [T (a), T (b)]. Thanks to (2.5), for any a, b, c, d ∈ V ,

2 2 [β(b), β(c)]T · β(a · d) + [[a, b]T , β(c)]T · β (d)+ β (b) · ([a, c]T · β(d)) − β2(a) · (β(b) · (c · d)) + β2(c) · (β(a) · (b · d)) 2 = ρ(α(T ([b, c]T ))ρ(α(T (a)))β(d)+ ρ(T ([[a, b]T , β(c)]T ))β (d) 2 2 + ρ(α (T (b)))ρ(T ([a, c]T ))β(d) − ρ(α (T (a)))ρ(α(T (b)))ρ(T (c))d + ρ(α2(T (c)))ρ(α(T (a)))ρ(T (b))d = ρ(α([T (b), T (c)]))ρ(α(T (a)))β(d)+ ρ([[T (a), T (b)], α(T (c))])β2 (d) + ρ(α2(T (b)))ρ([T (a), T (c)])β(d) − ρ(α2(T (a)))ρ(α(T (b)))ρ(T (c))d + ρ(α2(T (c)))ρ(α(T (a)))ρ(T (b))d = 0.

So, (V, ·, β) is a Hom-pre-Malcev algebra. The other statements follow immediately.

An obvious consequence of Proposition 3.2 is the construction of a Hom-pre-Malcev algebra in terms of a Rota-Baxter operator of weight zero of a Hom-Malcev algebra.

Corollary 3.1. Let R : A −→ A is a Rota-Baxter operator on a Hom-Malcev algebra (A, [−, −], α). Then (A, ·, α) is a Hom-pre-Malcev algebra, where for all x,y ∈ A,

x · y = [R(x),y].

Example 3.1. There is a four-dimensional Malcev algebra (A, [−, −]) with multiplication table [59, Example 3.1] for a basis {e1, e2, e3, e4}:

[−, −] e1 e2 e3 e4 e1 0 −e2 −e3 e4 e2 e2 0 2e4 0 e3 e3 −2e4 0 0 e4 −e4 0 0 0

Let R be the operator deﬁned, with respect to the basis {e1, e2, e3, e4}, by a R(e )= e + 4 e , R(e )= λ e , R(e )= R(e ) = 0, 1 1 2 4 2 1 3 3 4 where a4 and λ1 are parameters in K. By a direct computation, we can verify that R is a Rota-Baxter operator on A.

10 Now, using the previous corollary, there is a pre-Malcev algebra structure on A with the multiplication ” · ” given by x · y = [R(x),y], ∀x,y ∈ A, that is

· e1 e2 e3 e4 −a4 e1 2 e4 −e2 −e3 e4 e2 λ1e3 −2λ1e4 0 0 e3 0 0 00 e4 0 0 00 Using suitable algebra morphism α, we can twist the Malcev algebra A into Hom-Malcev algebras. With a bit of computation, one can check that one class of algebra morphisms α: A → A is given by

α(e1)= e1 + a4e4, α(e2)= −e2 + b3e3, α(e3)= −e3, α(e4)= −e4, where a4 and b3 are arbitrary scalars in K. There is a Hom-Malcev algebra Aα = (A, [−, −]α = α◦[−, −], α) with multiplication table:

[−, −]α e1 e2 e3 e4 e1 0 −α(e2) e3 −e4 e2 α(e2) 0 −2e4 0 e3 −e3 2e4 0 0 e4 e4 0 0 0

Then we can check that R is a Rota-Baxter operator on Hom-Malcev algebra Aα. There- fore, there exists a Hom-pre-Malcev algebra Aα = (A, ·α, α) where the multiplication ” ·α ” is given by x ·α y = α([R(x),y]), ∀x,y ∈ A, that is

·α e1 e2 e3 e4 a4 e1 2 e4 −α(e2) e3 −e4 e2 −λ1e3 2λ1e4 0 0 e3 0 0 00 e4 0 0 00

Example 3.2. There is a ﬁve-dimensional Malcev algebra (A, [−, −]) with multiplication table [59, Example 3.4] for a basis {e1, e2, e3, e4, e5}:

[−, −] e1 e2 e3 e4 e5 e1 0 0 0 e2 0 e2 0 0 00 e3 e3 0 0 000 e4 −e2 0 000 e5 0 −e3 0 0 0

11 Let R be the operator deﬁned, with respect to the basis {e1, e2, e3, e4, e5}, by −b R(e1)= e1 + a4e4 + a5e5, R(e2)= be3, R(e3)= R(e5) = 0, R(e4)= e2, a5 where b, a4 and a5 are parameters in K. By a direct computation, we can verify that R is a Rota-Baxter operator on A. Now, using the previous corollary, there is a pre-Malcev algebra structure ” · ” on A given by x · y = [R(x),y], ∀x,y ∈ A, that is · e1 e2 e3 e4 e5 e1 −a4e2 −a5e3 0 e2 0 e2 0 0 000 e3 0 0 000 −b e4 0 0 00 a5 e3 e5 0 0 000 Using suitable algebra morphism α given by

a4 α(e1)= e1, α(e2)= e2, α(e3)= e3, α(e4)= λ2e3 + e4, α(e5)= λ2e3 + e5, a5 where λ2, a4 and a5 are arbitrary scalars in K, we can twist the Malcev algebra A into Hom-Malcev algebra Aα = (A, [−, −]α = α ◦ [−, −], α) with multiplication table:

[−, −]α e1 e2 e3 e4 e5 e1 0 0 0 e2 0 e2 0 0 00 e3 e3 0 0 000 e4 −e2 0 000 e5 0 −e3 0 0 0 Then we can check that R is a Rota-Baxter operator on A. Therefore there exists a Hom-pre-Malcev algebras Aα = (A, ·α, α) with a multiplication ” ·α ” given by x ·α y = α([R(x),y]), ∀x,y ∈ A, that is

·α e1 e2 e3 e4 e5 e1 −a4e2 −a5e3 0 e2 0 e2 0 0 000 e3 0 0 000 −b e4 0 0 00 a5 e3 e5 0 0 000

12 Deﬁnition 3.4. A derivation of a Hom-Malcev algebra (A, [−, −], α) is a linear map D : A → A satisfying for all x,y ∈ A,

α ◦ D = D ◦ α, D[x,y] = [D(x),y] + [x, D(y)].

Proposition 3.3. Let (A, [−, −], α) be a Hom-Malcev algebra and R : A → A be a bijective linear map. Then R is a Rota-Baxter operator on A if and only if R−1 is a derivation on A.

Proof. Let (A, [−, −], α) be a Hom-Malcev algebra.Then, for any x,y ∈ A, R is a Rota- Baxter operator on A if and only if [R(x), R(y)] = R([R(x),y] + [x, R(y)]), which is equivalent to R−1([u, v]) = [u, R−1(v)] + [R−1(u), v], where u = R(x), v = R(y).

Hom-pre-Malcev algebras are related to Hom-pre-alternative algebras analogously to how Hom-pre-Lie algebras are related to Hom-dendriform algebras [43].

Deﬁnition 3.5 ( [66]). A Hom-pre-alternative algebra is a quadruple (A, ≺, ≻, α), where A is a vector space, ≺, ≻: A ⊗ A −→ A are linear maps and α ∈ gl(A) satisfying for all x,y,z ∈ A and x ∗ y = x ≺ y + x ≻ y,

(x ≻ y) ≺ α(z) − α(x) ≻ (y ≺ z) + (y ≺ x) ≺ α(z) − α(y) ≺ (x ∗ z) = 0, (3.5) (x ≻ y) ≺ α(z) − α(x) ≻ (y ≺ z) + (x ∗ z) ≻ α(y) − α(x) ≻ (z ≻ y) = 0, (3.6) (x ∗ y) ≻ α(z) − α(x) ≻ (y ≻ z) + (y ∗ x) ≻ α(z) − α(y) ≻ (x ≻ z) = 0, (3.7) (x ≺ y) ≺ α(z) − α(x) ≺ (y ∗ z) + (x ≺ z) ≺ α(y) − α(x) ≺ (z ∗ y) = 0. (3.8)

Proposition 3.4. Let (A, ≺, ≻, α) be a Hom-pre-alternative algebra. Then (A, ∗, α) is a Hom-alternative algebra

Now, we consider a Hom-alternative algebra (A, ∗, α), a vector space V and a linear map β : V → V . Recall that, a bimodule of A with respect to β is given by linear maps l, r : A → End(V ) satisfying the following conditions:

l(x2)β = l(α(x))l(x), (3.9) r(x2)β = r(α(x))r(x), (3.10) r(α(y))l(x) − l(α(x))r(y)= r(x ∗ y)β − r(α(y))r(x), (3.11) l(y ∗ x)β − l(α(y))l(x)= l(α(y))r(x) − r(α(x))l(y). (3.12)

13 Deﬁnition 3.6. An O-operator of Hom-alternative algebra (A, ∗, α) with respect to the bimodule (V, l, r, β) is a linear map T : V → A such that, for all a, b ∈ V ,

α ◦ T = T ◦ β and T (a) ∗ T (b)= T l(T (a))b + r(T (b))a . (3.13) Remark 3.1. Rota-Baxter operator of weight 0 on a Hom-alternative algebra (A, ∗, α) is just an O-operator associated to the bimodule (A,L,R,α), where L and R are the left and right multiplication operators corresponding to the multiplication ∗.

Proposition 3.5. With the above notations, the triplet (V, l − r, β) deﬁnes a module of the Hom-Malcev admissible algebra (A, [−, −], α), and T is an O-operator of (A, [−, −], α) with respect to (V, l − r, β).

Proof. First, note that (V, l, r, β) is a representation of a Hom-alternative algebra A if and only if the direct sum (A ⊕ V, ⋆, α + β) of vector spaces is a Hom-alternative algebra (the semi-direct product) by deﬁning multiplication in A ⊕ V by

(x + a) ⋆ (y + b)= x ∗ y + l(x)b + r(y)a, ∀x,y ∈ A, a, b ∈ V.

Next, for its associated Hom-Malcev admissible algebra (A ⊕ V, z[−}|, −{], α + β),

z[x + a,}| y + b{] =(x + a) ⋆ (y + b) − (y + b) ⋆ (x + a) =x ∗ y + l(x)b + r(y)a − y ∗ x − l(y)a − r(x)b =[x,y] + (l − r)(x)b − (l − r)(y)a.

According to Proposition 2.1, (V, l−r, β) is a representation of (A, [−, −], α). Moreover, T is an O-operator of (A, [−, −], α) with respect to (V, l − r, β) since

[T (a), T (b)] =T (a) ∗ T (b) − T (b) ∗ T (a) =T l(T (a))b + r(T (b))a − T l(T (b))a + r(T (a))b =T (l − r)(T (a))b − (l − r)(T (b))a . Theorem 3.2. Let T : V → A be an O-operator of Hom-alternative algebra (A, ∗, α) with respect to the bimodule (V, l, r, β). Then (V, ≺, ≻, β) be a Hom-pre-alternative algebra, where for all a, b ∈ V ,

a ≻ b = l(T (a))b and a ≺ b = r(T (b))a. (3.14)

Moreover, if (V, ·, β) is the Hom-pre-Malcev algebra associated to the Hom-Malcev admissible algebra (A, [−, −], α) on the module (V, l − r, β), then a · b = a ≻ b − b ≺ a.

14 Proof. For any a, b, c ∈ V , using (3.11) and (3.13) yields

(a ≻ b) ≺ β(c) − β(a) ≻ (b ≺ c) + (b ≺ a) ≺ β(c) − β(b) ≺ (a ∗ c) = r(T (β(c)))l(T (a))b − l(T (β(a)))r(T (c))b + r(T (β(c)))r(T (a))b − r(T (a ∗ c))β(b) = r(α(T (c)))l(T (a))b − l(α(T (a)))r(T (c))b + r(α(T (c)))r(T (a))b − r(T (a ∗ c))β(b) = 0.

The other identities for (V, ≺, ≻, β) being a Hom-pre-alternative algebras can be veriﬁed similarly. Moreover, using (3.2) and (3.14),

a · b = (l − r)(T (a))b = l(T (a))b − r(T (a))b = a ≻ b − b ≺ a.

Corollary 3.2. Let (A, ∗, α) be a Hom-alternative algebra and R : A → A be a Rota- Baxter operator of weight 0 such that Rα = αR. If multiplications ≺ and ≻ on A are deﬁned for all x,y ∈ A by x ≺ y = x ∗ R(y) and x ≻ y = R(x) ∗ y, then (A, ≺, ≻, α) is a Hom-pre-alternative algebra. Moreover, if (A, ·, α) be the Hom-pre-Malcev algebra associated to the Hom-Malcev admissible algebra (A, [−, −], α), then x · y = x ≻ y − y ≺ x.

Moreover, Hom-Malcev algebras, Hom-alternative algebras, Hom-pre-Malcev algebras and Hom-pre-alternative algebras are closely related as follows (in the sense of commutative diagram of categories):

· =≺−≻ Hom-pre-alt alg (A, ≺, ≻, α) // Hom-pre-Malcev alg (A, ·, α) O O ∗ =≺ + ≻ R-B Commutator R-B (3.15) Commutator Hom-alt alg (A, ∗, α) // Hom-Malcev alg (A, [−, −], α)

3.2 Bimodules and O-perators of Hom-pre-Malcev algebras In this subsection, we introduce and study bimodules of Hom-pre-Malcev algebras.

Deﬁnition 3.7. Let (A, ·, α) be a Hom-pre-Malcev algebra and V be a vector space. Let ℓ, r : A −→ End(V ) be two linear maps and β ∈ End(V ). Then (V,ℓ,r,β) is called

15 a bimodule of A if the following conditions hold for any x,y,z ∈ A:

βℓ(x)= ℓ(α(x))β, βr(x)= r(α(x))β, (3.16) r(α2(x))ρ(α(y))ρ(z) − r(α(z) · (y · x))β2 + ℓ(α2(y))r(z · x)β (3.17) + ℓ(α([y, z]))r(α(x))β − ℓ(α2(z))r(α(x))ρ(y) = 0, ℓ(α2(y))ℓ(α(z))r(x) − r(α2(x))ρ(α(y))ρ(z) − ℓ(α2(z))r(y · x)β (3.18) − r(α(z · x))ρ(α(y))β + r([z,y] · α(x))β2 = 0, r(α(y) · (z · x))β2 + r(α2(x))ρ([y, z])β − ℓ(α2(y))ℓ(α(z))r(x) (3.19) + r(α(y · x))ρ(α(z))β + ℓ(α2(z))r(α(x))ρ(y) = 0, ℓ([[x,y], α(z)])β2 − ℓ(α2(x))ℓ(α(y))ℓ(z)+ ℓ(α2(z))ℓ(α(x))ℓ(y) (3.20) + ℓ(α([y, z])ℓ(α(x))β + ℓ(α2(y))ℓ([x, z])β = 0,

where ρ(x)= ℓ(x) − r(x) and [x,y]= x · y − y · x.

Now, deﬁne a linear operation ·⋉ : ⊗2(A ⊕ V ) −→ (A ⊕ V ) by

(x + a) ·⋉ (y + b) := x · y + ℓ(x)(b)+ r(y)(a), ∀x,y ∈ A, a, b ∈ V,

and a linear map α + β : A ⊕ V −→ A ⊕ V by

(α + β)(x + a) := α(x)+ β(a), ∀x ∈ A, a ∈ V.

Proposition 3.6. With the above notations, (A ⊕ V, ·⋉, α + β) is a Hom-pre-Malcev ⋉α,β ⋉ algebra, which is denoted by A (ℓ, r) V or simply A V and called the semi-direct product of the Hom-pre-Malcev algebra (A, ·, α) and the representation (V,ℓ,r,β).

Proof. For any x,y,z,t ∈ A and a, b, c, d ∈ V ,

(α + β)[y + b, z + c] ·⋉ (α + β) (x + a) ·⋉ (t + d) ρ = α([y, z]) + ρ(α(y))β(c) − ρ(α(z))β(b) ·⋉ α(x · t)+ ℓ(α(x))d + r(α(t))a = α([y, z]) · α(x · t)+ ℓ(α([y, z])) ℓ(α(x))β(d)+ r(α(t))β(a) + r(α(x · t)) ρ(α(y))β(c) − ρ(α(z))β(b) , 2 2 [[x + a, y + b]ρ, (α + β)(z + c)]ρ ·⋉ (α + β )(t + d) 2 2 = [[x,y]+ ρ(x)b − ρ(y)a, α(z)+ β(c)]ρ ·⋉ (α (t)+ β (d)) 2 2 = [[x,y], α(z)] + ρ([x,y])β(c) − ρ(α(z))(ρ(x)b − ρ(y)a) ·⋉ (α (t)+ β (d)) = [[x,y], α(z)] · α2(t)+ ℓ([[x,y], α(z)])β2 (d) + r(α2(t)) ρ([x,y])β(c) − ρ(α(z))(ρ(x)b − ρ(y)a) , 2 2 (α + β )(y + b) ·⋉ [x + a, z + c] ·⋉ (α + β)(t + d) ρ 16 2 2 = (α (y)+ β (b)) ·⋉ ([x, z]+ ρ(x)c − ρ(z)a) ·⋉ (α(t)+ β(d)) 2 2 = (α (y)+ β (b)) ·⋉ [x, z] · α(t)+ ℓ([x, z])β(d)+ r(α(t))(ρ(x)c − ρ(z)a) = α2(y) · ([x, z] · α(t)) + ℓ(α2(y)) ℓ([x, z])β(d)+ r(α(t))(ρ(x)c − ρ(z)a) + r [x, z] · α(t) β2(b), 2 2 (α + β )(x + a) ·⋉ (α + β)(y + b) ·⋉ ((z + c) ·⋉ (t + d)) 2 2 = (α (x)+ β (a)) ·⋉ (α(y)+ β(b)) ·⋉ ((z · t)+ ℓ(z)d + r(t)c) 2 2 = (α (x)+ β (a)) ·⋉ α(y) · (z · t)+ ℓ(α(y))(ℓ(z)d + r(t)c)+ r(z · t)β(b) = α2(x) · (α(y) · (z · t)) + ℓ(α2(x)) ℓ(α(y))(ℓ(z)d + r(t)c)+ r(z · t)β(b) + r α(y) · (z · t) β2(a), 2 2 (α + β )(z + c) ·⋉ (α + β)(x + a) ·⋉ ((y + b) ·⋉ (t + d)) 2 2 = (α (z)+ β (c)) ·⋉ (α(x)+ β(a)) ·⋉ ((y · t)+ ℓ(y)d + r(t)b) 2 2 = (α (z)+ β (c)) ·⋉ α(x) · (y · t)+ ℓ(α(x))(ℓ(y)d + r(t)b)+ r(y · t)β(a) = α2(z) · (α(x) · (y · t)) + ℓ(α2(z)) ℓ(α(x))(ℓ(y)d + r(t)b)+ r(y · t)β(a) + r α(x) · (y · t) β2(c).

Hence (A ⊕ V, ·⋉, α + β) is a Hom-pre-Malcev algebra if and only if (V,ℓ,r,β) is a bimodule of (A, ·, α).

Proposition 3.7. Let (V,ℓ,r,β) be a bimodule of a Hom-pre-Malcev algebra (A, ·, α) and (A, [−, −], α) be its associated Hom-Malcev algebra. Then, (i) (V,ℓ,β) is a representation of (A, [−, −], α), (ii) (V,ℓ − r, β) is a representation of (A, [−, −], α).

Proof. (i) The statement (i) follows immediately from (3.17). (ii) By Proposition 3.6, A ⋉α,β V is a Hom-pre-Malcev algebra. For its associated ℓ,r Hom-Malcev algebra (A ⊕ V, z[−}|, −{], α + β),

z[x + a,}| y + b{] = (x + a) ·⋉ (y + b) − (y + b) ·⋉ (x + a) = x · y + ℓ(x)b + r(y)a − y · x − ℓ(y)a − r(x)b = [x,y] + (ℓ − r)(x)b − (ℓ − r)(y)a.

By Proposition 2.1, (V,ℓ − r, β) is a representation of (A, [−, −], α).

The following result gives a construction of a bimodule of Hom-pre-Malcev algebra starting with a classical one by means of the Yau twist procedure.

17 Proposition 3.8. Let (A, ·) be a pre-Malcev algebra, α : A → A be a morphism on A, (V,ℓ,r) be a bimodule of (A, ·) and β : V → V be a linear map such that

βℓ(x)(b)= ℓ(α(x))(β(b)) and βr(x)(b)= r(α(x))(β(b)).

Then (V, ℓ, r, β) is a bimodule of (A, ·α, α), where e e ℓ(x)(b)= ℓ(α(x))(β(b)) and r(x)(b)= r(α(x))(β(b)). e e Proof. For x,y,z ∈ A, b ∈ V ,

βℓ(x)(b)= β(ℓ(α(x))(β(b))) = ℓ(α2(x))β2(b)= ℓ(α(x))β(b), e e βr(x)(b)= β(r(α(x))(β(b))) = r(α2(x))β2(b)= r(α(x))β(b), e e that is (3.16) holds for ℓ and r. Using (3.16) and (3.17), we get: e e 2 2 2 r(α (x))ρ(α(y))ρ(z)(b) − r(α(z) ·α (y ·α x))β (b)+ ℓ(α (y))r(z ·α x)β(b) e e e e 2 e e + ℓ(α([y, z]α))r(α(x))β(b) − ℓ(α (z))r(α(x))ρ(y)(b)= 3 e e e e e β r(x)ρ(y)ρ(z) − r(z · (y · x)) + ℓ(y)r(z · x)β + ℓ([y, z])r(x) − ℓ(z)r(x)ρ(y)(b) = 0.

Similarly, (3.18)-(3.20) also hold for (V, ℓ, r, β). e e If (A, ·, α) is a Hom-pre-Malcev algebra and (A, [−, −], α) is the associated Hom- Malcev algebra, then (A, L·, α) is a representation of (A, [−, −], α) , where L· is the left operation of (A, ·, α) given by L(x)(y)= x · y. Proposition 3.9. Let (A, ·, α) be a Hom-algebra. Then (A, ·, α) is a Hom-pre-Malcev algebra if and only if (A, [−, −], α) deﬁned by (3.3) is a Hom-Malcev algebra and (A, L·, α) is a representation of (A, ·, α).

Proof. It follows from the deﬁnition of Hom-Malcev algebra and representation of Hom- Malcev algebra. Then, for any x,y,z,t ∈ A,

α([y, z]) · α(x · t) + [[x,y], α(z)] · α2(t)+ α2(y) · ([x, z] · α(t)) − α2(x) · (α(y) · (z · t)) + α2(z) · (α(x) · (y · t)) 2 2 = L·(α[y, z])L·(α(x))α + L·([[x,y], α(z)])α + L·(α (y))L·([x, z])α 2 2 − L·(α (x))L·(α(y))L·(z)+ L·(α (z))L·(α(x))L·(x)(t) = 0.

As in [4,25], we rephrase the deﬁnition of O-operator in terms of Hom-pre-Malcev algebras as follows.

18 Deﬁnition 3.8. Let (A, ·, α) be a Hom-pre-Malcev algebra and (V,ℓ,r,β) be a bimodule. A linear map T : V → A is called an O-operator associated to (V,ℓ,r,β) if T satisﬁes

T ◦ β = α ◦ T, (3.21) T (a) · T (b)= T ℓ(T (a))b + r(T (b))a , ∀a, b ∈ V. (3.22) Remark 3.2. Let T is an O-operator of a Hom-pre-Malcev algebra (A, ·, α) associated to (V,ℓ,r,β). Then T is an O-operator of its associated Hom-Malcev algebra (A, [−, −], α) associated to (V,ℓ − r, β).

Proof. By Proposition 3.7, for all a, b ∈ V,

[T (a), T (b)] = T (a) · T (b) − T (b) · T (a) = T (ℓ(T (a))b + r(T (b))a) − T (ℓ(T (b))a + r(T (a))b) = T ((ℓ − r)(T (a))b − (ℓ − r)(T (b))a).

4 Hom-M-dendriform algebras

The goal of this section is to introduce the notion of Hom-M-dendriform algebras which is the Hom-type of M-dendriform and show that is a Hom-pre-Malcev.

Deﬁnition 4.1. Hom-M-dendriform algebra is a vector space A endowed with two bilinear products ◮, ◭: A × A → A and a linear map α : A → A such that for all x,y,z,t ∈ A and

x · y = x ◭ y + x ◮ y, (4.1) x ⋄ y = x ◭ y − y ◮ x, (4.2) [x,y]= x · y − y · x = x ⋄ y − y ⋄ x, (4.3) the following identities are satisﬁed:

(α(z) ⋄ (y ⋄ x)) ◮ α2(t) − α2(x) ◮ (α(y) · (z · t)) + α2(z) ◭ (α(x) ◮ (y · t)) (4.4) +α([y, z]) ◭ α(x ◮ t) − α2(y) ◭ ((z ⋄ x) ◮ α(t)) = 0, α2(z) ◭ (α(x) ◭ (y ◮ t)) − (α(z) ⋄ (x ⋄ y)) ◮ α2(t) − α2(x) ◭ (α(y) ◮ (z · t)) (4.5) −α(z ⋄ y) ◮ α(x · t)+ α2(y) ◮ ([x, z] · α(t)) = 0, α2(z) ◭ (α(x) ◭ (y ◭ t)) + ([x,y] ⋄ α(z)) ◮ α2(t) − α2(x) ◭ (α(y) ◭ (z ◮ t)) (4.6) +α(y ⋄ z) ◮ α(x · t)+ α2(y) ◭ ((x ⋄ z) ◮ α(t)) = 0, [[x,y], α(z)] ◭ α2(t) − α2(x) ◭ (α(y) ◭ (z ◭ t)) + α2(z) ◭ (α(x) ◭ (y ◭ t)) (4.7) +α([y, z]) ◭ α(x ◭ t)+ α2(y) ◭ ([x, z] ◭ α(t)) = 0.

19 Now, we provide a way to construct Hom-M-dendriform algebras starting from an M-dendriform algebra and an algebra endomorphism.

Proposition 4.1. Let (A, ◮, ◭) be an M-dendriform algebra and α : A → A be an algebra morphism. Then (A, ◮α, ◭α, α) is a Hom-M-dendriform algebra where

x ◮α y = α(x ◮ y) and x ◭α y = α(x ◭ y).

Proof. We only prove that (A, ◮α, ◭α, α) satisfies the first Hom-M-dendriform identity. The other identities for (A, ◮α, ◭α, α) being a Hom-M-dendriform algebra can be verified similarly. In fact, for any x,y,z,t ∈ A,

2 2 2 (α(z) ⋄α (y ⋄α x)) ◮α α (t) − α (x) ◮α (α(y) ·α (z ·α t)) + α (z) ◭α (α(x) ◮α (y ·α t)) 2 + α([y, z]α) ◭α α(x ◮α t) − α (y) ◭α ((z ⋄α x) ◮α α(t)) 3 ◮ ◮ ◭ ◮ ◭ ◮ = α (z ⋄ (y ⋄ x)) t − x (y · (z · t)) + z (x (y · t)) + [y, z] (x t) ◭ ◮ − y ((z ⋄ x) t) = 0.

Hence (A, ◮α, ◭α, α) is a Hom-M-dendriform algebra.

Theorem 4.1. Let (A, ◮, ◭, α) be a Hom-M-dendriform algebra. (i) The product given by (4.1) defines a Hom-pre-Malcev algebra (A, ·, α), called the associated horizontal Hom-pre-Malcev algebras. (ii) The product given by (4.2) defines a Hom-pre-Malcev algebra (A, ⋄, α), called the associated vertical Hom-pre-Malcev algebras. (iii) the associated horizontal and vertical Hom-pre-Malcev algebras (A, ·, α) and (A, ⋄, α) of a Hom-M-dendriform algebra (A, ◮, ◭, α) have the same associated Hom-Malcev algebras (A, [−, −], α) defined by (4.3), called the associated Hom- Malcev algebra of the Hom-M-dendriform algebra (A, ◮, ◭, α).

Proof. We will just prove (i). In fact, using (3.1), for any x,y,z,t ∈ A,

α([y, z]) · α(x · t) + [[x,y], α(z)] · α2(t)+ α2(y) · ([x, z] · α(t)) − α2(x) · (α(y) · (z · t)) + α2(z) · (α(x) · (y · t)) = α([y, z]) ◭ α(x ◭ t)+ α([y, z]) ◭ α(x ◮ t)+ α([y, z]) ◮ α(x ◭ t) + α([y, z]) ◮ α(x ◮ t) + [[x,y], α(z)] ◭ α2(t) + [[x,y], α(z)] ◮ α2(t) + α2(y) ◭ ([x, z] ◭ α(t)) + α2(y) ◭ ([x, z] ◮ α(t)) + α2(y) ◮ ([x, z] ◭ α(t)) + α2(y) ◮ ([x, z] ◮ α(t)) + α2(z) ◭ (α(x) ◭ (y ◭ t)) + α2(z) ◭ (α(x) ◭ (y ◮ t))

20 + α2(z) ◭ (α(x) ◮ (y ◭ t)) + α2(z) ◭ (α(x) ◮ (y ◮ t)) + α2(z) ◮ (α(x) ◭ (y ◭ t)) + α2(z) ◮ (α(x) ◭ (y ◮ t)) + α2(z) ◮ (α(x) ◮ (y ◭ t)) + α2(z) ◮ (α(x) ◮ (y ◮ t)) − α2(x) ◭ (α(y) ◭ (z ◭ t)) − α2(x) ◭ (α(y) ◭ (z ◮ t)) − α2(x) ◭ (α(y) ◮ (z ◭ t)) − α2(x) ◭ (α(y) ◮ (z ◮ t)) − α2(x) ◮ (α(y) ◭ (z ◭ t)) − α2(x) ◮ (α(y) ◭ (z ◮ t)) − α2(x) ◮ (α(y) ◮ (z ◭ t)) − α2(x) ◮ (α(y) ◮ (z ◮ t)) = α([y, z]) ◭ α(x ◮ t) + (α(z) ⋄ (y ⋄ x)) ◮ α2(t) − α2(y) ◭ ((z ⋄ x) ◮ α(t)) + α2(z) ◭ (α(x) ◮ (y · t)) − α2(x) ◮ (α(y) · (z · t)) − α(z ⋄ y) ◮ α(x · t) − (α(z) ⋄ (x ⋄ y)) ◮ α2(t)+ α2(y) ◮ ([x, z] · α(t)) + α2(z) ◭ (α(x) ◭ (y ◮ t)) − α2(x) ◭ (α(y) ◮ (z · t)) + α(y ⋄ z) ◮ α(x · t) + ([x,y] ⋄ α(z)) ◮ α2(t) + α2(y) ◭ ((x ⋄ z) ◮ α(t)) + α2(z) ◮ (α(x) · (y · t)) − α2(x) ◭ (α(y) ◭ (z ◮ t)) + [[x,y], α(z)] ◭ α2(t)+ α([y, z]) ◭ α(x ◭ t)+ α2(y) ◭ ([x, z] ◭ α(t)) + α2(z) ◭ (α(x) ◭ (y ◭ t)) − α2(x) ◭ (α(y) ◭ (z ◭ t)) = 0. Hom-M-dendriform algebras are closely related to bimodules for Hom-pre-Malcev algebras.

Proposition 4.2. Let (A, ◮, ◭, α) be a Hom-M-dendriform algebra. Let L◭ and R◮ be the left and right multiplication operators corresponding respectively to the two operations ◮ and ◭. Then (A, L◭, R◮, α) is a bimodule of its associated horizontal Hom-pre- Malcev algebra (A, ·, α).

Proof. We verify that (3.17) and (3.20) hold for (A, L◭, R◮, α). For any x,y,z,t ∈ A, 2 2 2 R◮(α (x))L⋄(α(y))L⋄(z)(t) − R◮(α(z) · (y · x))α (t)+ L◭(α (y))R◮(z · x)α(t) 2 + L◭(α([y, z]))R◮(α(x))α(t) − L◭(α (z))R◮(α(x))L⋄(y)(t) = (α(y) ⋄ (z ⋄ t)) ◮ α2(x) − α2(t) ◮ (α(z) · (y · x)) + α2(y) ◭ (α(t) ◮ (z · x)) + α([y, z]) ◭ α(x ◮ t) − α2(z) ◭ ((y ⋄ t) ◮ α(x)) = 0. 2 2 2 L◭[[x,y], α(z)]α (t) − L◭(α (x))L◭(α(y))L◭(z)(t)+ L◭(α (z))L◭(α(x))L◭(y)(t) 2 + L◭(α([y, z]))L◭(α(x))α(t)+ L◭(α (y))L◭([x, z])α(t) = [[x,y], α(z)] ◭ α2(t) − α2(x) ◭ (α(y) ◭ (z ◭ t)) + α2(z) ◭ (α(x) ◭ (y ◭ t)) + α([y, z]) ◭ α(x ◭ t)+ α2(y) ◭ ([x, z] ◭ α(t)) = 0.

Other identities can be proved using similar computations. Thus, (A, L◭, R◮, α) is a representation of Hom-pre-Malcev (A, ·, α). Proposition 4.3. Let (A, ◮, ◭, α) be a Hom-M-dendriform algebra. With two binary operations ◮t, ◭t: A ⊗ A → A deﬁned for all x,y ∈ A by x ◮t y = −y ◮ x,x ◭t y = x ◭ y, (4.8)

21 (A, ◮t, ◭t, α) is a Hom-M-dendriform algebra. Its associated horizontal Hom-pre-Malcev algebra is the associated vertical Hom-pre-Malcev algebra (A, ⋄, α) of (A, ◮, ◭, α), and its associated vertical Hom-pre-Malcev algebra is the associated horizontal Hom-pre- Malcev algebra (A, ·, α) of (A, ◮, ◭, α), that is, x ·t y = x ◭t y + x ◮t y = x ◭ y − y ◮ y = x ⋄ y, (4.9) x ⋄t y = x ◭t y − y ◮t x = x ◭ y + x ◮ y = x · y, (4.10) [x,y]t = x ◭t y + x ◮t y − y ◭t x − y ◮t x (4.11) = x ◭ y − y ◮ x − y ◭ x + x ◮ y = [x,y]. Proof. By (4.9)-(4.11), for all x,y,z,t ∈ A, [[x,y]t, α(z)]t ◭t α2(t) − α2(x) ◭t (α(y) ◭t (z ◭t t)) + α2(z) ◭t (α(x) ◭t (y ◭t t)) + α([y, z]t) ◭t α(x ◭t t)+ α2(y) ◭t ([x, z]t ◭t α(t)) = [[x,y], α(z)] ◭ α2(t) − α2(x) ◭ (α(y) ◭ (z ◭ t)) + α2(z) ◭ (α(x) ◭ (y ◭ t)) + α([y, z]) ◭ α(x ◭ t)+ α2(y) ◭ ([x, z] ◭ α(t)). Similarly, (4.4)t=(4.4), (4.5)t=(4.5) and (4.6)t=(4.6). Therefore, (A, ◮t, ◭t, α) is a Hom-M-dendriform algebra. Deﬁnition 4.2. Let (A, ◮, ◭, α) be a Hom-M-dendriform algebra. The Hom-M-dendriform algebra (A, ◮t, ◭t, α) given by (4.8) is called the transpose of (A, ◮, ◭, α). Examples of Hom-M-dendriform algebras can be constructed from Hom-pre-Malcev algebras with O-operators. For brevity, we only give the study involving the associated horizontal Hom-pre-Malcev algebras. Proposition 4.4. Let (A, ·, α) be a Hom-pre-Malcev algebra and (V,ℓ,r,β) be a bimodule of A. Let T be an O-operator of (A, ·, α) associated to (V,ℓ,r,β). Then (V, ◮, ◭, β) is a Hom-M-dendriform algebra, where a ◮ b = r(T (b))a, a ◭ b = ℓ(T (a))b, ∀a, b ∈ V. (4.12) Therefore, there is a Hom-pre-Malcev algebra on V given in Theorem 4.1 as the associated horizontal Hom-pre-Malcev algebra of (V, ◮, ◭, β), and T is a morphism of Hom- pre-Malcev algebras. Moreover, T (V ) = {T (v)/ v ∈ V } ⊂ A is a Hom-pre-Malcev subalgebra of (A, ·, α), and there is an induced Hom-M-dendriform on (T (V ), ⊲, ⊳, α) given by T (a) ⊲ T (b)= T (a ◮ b), T (a) ⊳ T (b)= T (a ◭ b), ∀a, b ∈ V. Its corresponding associated horizontal Hom-pre-Malcev algebraic structure on T (V ) is just the subalgebra of the Hom-pre-Malcev (A, ·, α), and T is a homomorphism of Hom-M-dendriform algebras.

22 Proof. For any a, b, c, d ∈ V , using (3.17) and Deﬁnition 3.8, (β(c) ⋄ (b ⋄ a)) ◮ β2(d) − β2(a) ◮ (β(b) · (c · d)) + β2(c) ◭ (β(a) ◮ (b · d)) + β([b, c]) ◭ β(a ◮ d) − β2(b) ◭ ((c ⋄ a) ◮ β(d)) = (β(c) ◭ (b ◭ a)) ◮ β2(d) − ((b ◭ a) ◮ β(c)) ◮ β2(d) − (β(c) ◭ (a ◮ b)) ◮ β2(d) + ((a ◮ b) ◮ β(c)) ◮ β2(d) − β2(a) ◮ (β(b) ◭ (c ◭ d)) − β2(a) ◮ (β(b) ◮ (c ◭ d)) − β2(a) ◮ (β(b) ◭ (c ◮ d)) − β2(a) ◮ (β(b) ◮ (c ◮ d)) + β2(c) ◭ (β(a) ◮ (b ◭ d)) + β2(c) ◭ (α(a) ◮ (b ◮ d)) + β(b ◭ c) ◭ β(a ◮ d)+ β(b ◮ c) ◭ β(a ◮ d) − β(c ◭ b) ◭ β(a ◮ d) − β(c ◮ b) ◭ β(a ◮ d) − β2(b) ◭ ((c ◭ a) ◮ β(d)) + β2(b) ◭ ((a ◮ c) ◮ β(d)) = r(α2(T (d)))ℓ(α(T (c)))ℓ(T (b))a − r(α2(T (d)))r(α(T (c)))ℓ(T (b))a − r(α2(T (d)))ℓ(α(T (c)))r(T (b))a + r(α2(T (d)))r(α(T (c)))r(T (b))a − r(T (ℓ(T (β(b)))ℓ(T (c))d))β2 (a) − r(T (r(T (ℓ(T (c))d))β(b)))β2 (a) − r(T (ℓ(T (β(b)))r(T (d))c))β2 (a) − r(T (r(T (r(T (d))c))β(b)))β2(a) + ℓ(α2(T (c)))r(T (ℓ(T (b))d))β(a)+ ℓ(α2(T (c)))r(T (r(T (d))b))β(a) + ℓ(α(T (ℓ(T (b))c)))r(α(T (d))a)+ ℓ(α(T (r(T (c))b)))r(α(T (d))a) − ℓ(α(T (ℓ(T (c))b)))r(α(T (d))a) − ℓ(α(T (r(T (b))c)))r(α(T (d))a) − ℓ(α2(T (b)))r(α(T (d)))ℓ(T (c))a + ℓ(α2(T (b)))r(α(T (d)))r(T (c))a = r(α2(T (d)))ℓ(α(T (c)))ρ(T (b))a − r(α2(T (d)))r(α(T (c)))ρ(T (b))a − r(T (ℓ(T (β(b)))(T (c) · T (d))))β2(a) − r(T (r(T (c) · T (d))β(b)))β2(a) + ℓ(α2(T (c)))r(T (b) · T (d))β(a)+ ℓ(α(T (b) · T (c)))r(α(T (d))a) − ℓ(α(T (c) · T (b)))r(α(T (d))a) − ℓ(α2(T (b)))r(α(T (d)))ρ(T (c))a = r(α2(T (d)))ρ(α(T (c)))ρ(T (b))a − r(α(T (b)) · (T (c) · T (d)))β2(a) + ℓ(α2(T (c)))r(T (b) · T (d))β(a)+ ℓ(α([T (b), T (c)]))r(α(T (d))a) − ℓ(α2(T (b)))r(α(T (d)))ρ(T (c))a = 0. Using a similar computation, (4.5)-(4.7) can be checked. Therefore, (V, ◮, ◭, β) is a Hom-M-dendriform algebra. The other conclusions follow easily.

Now, we introduce the following concept of Rota-Baxter operator on a Hom-pre- Malcev algebra which is a particular case of O-operator. Deﬁnition 4.3. Let (A, ·, α) be a Hom-pre-Malcev algebra. A linear map R : A −→ A is called a Rota-Baxter operator of weight zero on A if R ◦ α = α ◦ R, and R(x) · R(y)= R R(x) · y + x · R(y) , ∀x,y ∈ A. 23 Corollary 4.1. Let (A, ·, α) be a Hom-pre-Malcev algebra and R : A → A be a Rota- Baxter operator of weight 0 for A. Deﬁne new operations on A by x ◮ y = x · R(y) and x ◭ y = R(x) · y. Then (A, ◮, ◭, α) is a Hom-M-dendriform algebra.

Lemma 4.1. Let R1 and R2 be two commuting Rota-Baxter operators (of weight zero) on a Hom-Malcev algebra (A, [−, −], α). Then R2 is a Rota-Baxter operator (of weight zero) on the Hom-pre-Malcev algebra (A, ·, α), where

x · y = [R1(x),y], ∀x,y ∈ A. Proof. For any x,y ∈ A,

R2(x) · R2(y) =[R1(R2(x)), R2(y)]

=R2([R1(R2(x)),y] + [R1(x), R2(y)])

=R2(R2(x) · y + x · R2(y)).

Corollary 4.2. Let R1 and R2 be two commuting Rota-Baxter operators (of weight zero) on a Hom-Malcev algebra (A, [−, −], α). Then there exists a Hom-M-dendriform algebra structure on A given by

x ◮ y = [R1(x), R2(y)], x ◭ y = [R1(R2(x)),y], ∀x,y ∈ A. (4.13)

Proof. By Lemma 4.1, R2 is a Rota-Baxter operator of weight zero on (A, ·, α), where

x · y = [R1(x),y]. Then, applying Corollary 4.1, there exists a Hom-M-dendriform algebraic structure on A given by

x ◮ y = x · R2(y) = [R1(x), R2(y)],

x ◭ y = R2(x) · y = [R1(R2(x)),y], ∀x,y ∈ A. Example 4.1. Consider the 4-dimensional Hom-Malcev algebra and the Rota-Baxter operators R in Example 3.1. Then, there is a Hom-M-dendriform algebraic structure on A given by 2 x ◮α y = α([R(x), R(y)]), x ◭α y = α([R (x),y]), ∀x,y ∈ A, that is

◮α e1 e2 e3 e4 ◭α e1 e2 e3 e4 a4 e1 0 λ1e3 0 0 e1 2 e4 −α(e2) e3 −e4 e2 −λ1e3 0 0 0 e2 0 0 00 . e3 0 0 00 e3 0 0 00 e4 0 0 00 e4 0 0 00

24 Example 4.2. Consider the 5-dimensional Hom-Malcev algebra and Rota-Baxter operators R in Example 3.2. Then, there is a Hom-M-dendriform algebraic structure on A given by 2 x ◮α y = α([R(x), R(y)]), x ◭α y = α([R (x),y]), ∀x,y ∈ A, that is ◭ ◮α e1 e2 e3 e4 e5 α e1 e2 e3 e4 e5 −ba4 e1 0 0 0 be3 0 e1 −a4e2 −a5e3 0 e2 a5 e3 e 0 0000 e 0 0 00 0 2 2 . e3 0 0000 e3 0 0 00 0 e4 −be3 00 0 0 e4 0 0 00 0 e5 0 0000 e5 0 0 00 0 Hom-M-dendriform algebras are related to Hom-alternative quadri-algebras in the same way Hom-L-dendriform algebras are related to Hom-quadri-algebras( see [14] for more details). Deﬁnition 4.4. Hom-alternative quadri-algebra is a 6-tuple (A, տ, ւ, ր, ց, α) consisting of a vector space A, four bilinear maps տ, ւ, ր, ց: A×A → A and a linear map α : A → A which is algebra morphism such that the following axioms are satisﬁed for all x,y,z ∈ A : r m r r ((x,y,z))α + ((y,x,z))α = 0, ((x,y,z))α + ((x,z,y))α = 0, n w n ne ((x,y,z))α + ((y,x,z))α = 0, ((x,y,z))α + ((x,z,y))α = 0, ne e w sw ((x,y,z))α + ((y,x,z))α = 0, ((x,y,z))α + ((x,z,y))α = 0, sw s m ℓ ((x,y,z))α + ((y,x,z))α = 0, ((x,y,z))α + ((x,z,y))α = 0, ℓ ℓ ((x,y,z))α + ((y,x,z))α = 0, where r ((x,y,z))α = (x տ y) տ α(z) − α(x) տ (y ∗ z) (right associator) ℓ ((x,y,z))α = (x ∗ y) ց α(z) − α(x) ց (y ց z) (left associator) m ((x,y,z))α = (x ց y) տ α(z) − α(x) ց (y տ z) (middle associator) n ((x,y,z))α = (x ր y) տ α(z) − α(x) ր (y ≺ z) (north associator) w ((x,y,z))α = (x ւ y) տ α(z) − α(x) ւ (y ∧ z) (west associator) s ((x,y,z))α = (x ≻ y) ւ α(z) − α(x) ց (y ւ z) (south associator) e ((x,y,z))α = (x ∨ y) ր α(z) − α(x) ց (y ր z) (east associator) ne ((x,y,z))α = (x ∧ y) ր α(z) − α(x) ր (y ≻ z) (north-east associator) sw ((x,y,z))α = (x ≺ y) ւ α(z) − α(x) ւ (y ∨ z) (south-west associator)

25 x ≻ y = x ր y + x ց y, x ≺ y = x տ y + x ւ y, x ∨ y = x ց y + x ւ y, x ∧ y = x ր y + x տ y, x ∗ y = x ≻ y + x ≺ y = x ց y + x ր y + x տ y + x ւ y.

Lemma 4.2. Let (A, ր, ց, ւ, տ, α) be some Hom-alternative quadri-algebra. Then (A, ≺, ≻, α) and (A, ∨, ∧, α) are Hom-pre-alternative algebras (called respectively horizontal and vertical Hom-pre-alternative structures associated to A), and (A, ∗, α) is a Hom-alternative algebra.

Deﬁnition 4.5 introduces the notion of bimodule of Hom-pre-alternative algebras.

Deﬁnition 4.5. Let (A ≺, ≻, α) be a Hom-pre-alternative algebra and (V, β) a vector space. Let L≻, R≻,L≺, , R≺ : A → gl(V ) be linear maps. Then, (V,L≻, R≻,L≺, , R≺, β) ia called a representation or a bimodule of (A ≺, ≻, α) if for any x,y ∈ A,

L≻(x ∗ y + y ∗ x)β = L≻(α(x))L≻(y)+ L≻(α(y))L≻(x), (4.14)

R≻(α(y))(L(x)+ R(x)) = L≻(α(x))R≻(y)+ R≻(x ≻ y)β, (4.15)

R≺(α(y))L≻(x)+ R≺(α(y))R≺(x)= L≻(α(x))R≺(y)+ R≺(x ∗ y)β, (4.16)

R≺(α(y))R≻(x)+ R≻(α(y))L≺(x)= L≺(α(x))R(y)+ R≻(x ∗ y)β, (4.17)

L≺(y ≺ x)β + L≺(x ≻ y)β = L≺(α(y))L(x)+ L≻(α(y))L≻(x), (4.18)

R≺(α(x))L≻(y)+ L≻(y ≻ x)β = L≻(y)R≺(x)+ L≻(α(y))L≻(x), (4.19)

R≺(α(x))R≻(y)+ R≻(α(y))R(x)= R≻(y ≺ x)β + R≻(x ≻ y)β, (4.20)

L≺(y ≻ x)β + R≻(α(x))L(y)= L≻(α(y))L≺(x)+ L≻(α(y))R≻(y), (4.21)

R≺(α(x))R≺(y)+ R≺(α(y))R≺(x)= R≺(x ∗ y + y ∗ x)β, (4.22)

R≺(α(y))L≺(x)+ L≺(x ≺ y)β = L≺(α(x))(R(y)+ L(y)), (4.23) where x ∗ y = x ≺ y + y ≻ x, L = L≺ + L≻ and R = R≺ + R≻.

Proposition 4.5. A tuple (V,L≻, R≻,L≺, R≺, β) is a bimodule of a Hom-pre-alternative algebra (A, ≺, ≻, α) if and only if the direct sum (A ⊕ V, ≪, ≫, α + β) is a Hom-pre-alternative algebra, where for any x,y ∈ A, a, b ∈ V ,

(x + a) ≪ (y + b)= x ≺ y + L≺(x)b + R≺(y)a,

(x + a) ≫ (y + b)= x ≻ y + L≻(x)b + R≻(x)a, (α ⊕ β)(x + a)= α(x)+ β(a).

⋉α,β ⋉ We denote it by A L≻,R≻,L≺,R≺ V or simply A V . Deﬁnition 4.6. Let (A, ≺, ≻, α) be a Hom-pre-alternative algebra. Then a linear map T : V → A is called an O-operator of (A, ≺, ≻, α) associated to a bimodule

26 (V,L≺, R≺,L≻, R≻, β) if T satisﬁes, for all a, b ∈ V ,

T ◦ β = α ◦ T,

T (a) ≻ T (b)= T L≻(T (a))b + R≻(T (b))a , (4.24) T (a) ≺ T (b)= T L≺(T (a))b + R≺(T (b))a . Proposition 4.6. Let (A, ≺, ≻, α) be some Hom-pre-alternative algebra, and let T be an O-operator of (A, ≺, ≻, α) associated to a bimodule (V,L≺, R≺,L≻, R≻, β). With products deﬁned, for any a, b ∈ V , by

a ց b = L≻(T (a))b, a ր b = R≻(T (b))a, (4.25) a ւ b = L≺(T (a))b, a տ b = R≺(T (b))a,

(V, ց, ր, ւ, տ, β) is a Hom-alternative quadri-algebra.

Proof. Set L = L≺ + L≻ and R = R≺ + R≻. For any a, b, c ∈ V ,

(a ∗ b) ց β(c) − β(a) ց (b ց c)

= (L(T (a))b + R(T (b))a) ց β(c) − β(a) ց (L≻(T (b))c)

= L≻(T (L(T (a))b + R(T (b))a))β(c) − L≻(α(T (a)))L≻(T (b))c

= L≻(T (a) ∗ T (b))β(c) − L≻(α(T (a)))L≻(T (b))c = 0. (a ւ b) տ β(c) − φ(a) ւ (b տ c + b ր c)

= R≺(α(T (c)))L≺(T (a))b − L≺(α(T (a)))(R≺(T (c))b + R≻(T (c))b)

= R≺(α(T (c)))L≺(T (a))b − L≺(α(T (a)))R(T (c))b, (a տ c + a ւ c) ւ β(b) − β(a) ւ (c ց b + c ւ b)

= L≺(T (R≺(T (c))a + L≺(T (a))c))β(b) − L≺(α(T (a)))(L≻(T (c))b + L≺(T (c))b)

= L≺(T (a) ≺ T (c))β(b) − L≺(α(T (a)))(L(T (c))b.

This means that

w sw ((a, b, c))β + ((a, c, b))β =R≺(α(T (c)))L≺(T (a))b − L≺(α(T (a)))R(T (c))b + L≺(T (a) ≺ T (c))β(b) − L≺(α(T (a)))L(T (c))b = 0,

since (V,L≺, R≺,L≻, R≻, β) is a bimodule of (A, ≺, ≻, α). The rest of identities can be proved using analogous computations.

Analogously to what happens for quadri-algebras [1], Rota-Baxter operators allow diﬀerent constructions for alternative quadri-algebras which is a particular case of O- operator.

27 Deﬁnition 4.7. Let (A, ≺, ≻, α) be a Hom-pre-alternative algebra. A Rota-Baxter operator of weight 0 on A is a linear map R : A → A such that Rα = αR and the following conditions are satisﬁed, for all x,y ∈ A:

R(x) ≻ R(y)= R(x ≻ R(y)+ R(x) ≻ y), (4.26) R(x) ≺ R(y)= R(x ≺ R(y)+ R(x) ≺ y). (4.27)

Corollary 4.3. Let (A, ≺, ≻, α) be a Hom-pre-alternative algebra and R : A → A be a Rota-Baxter operator of weight 0 for A. Then (A, ր, ց, ւ, տ, α) is a Hom-alternative quadri-algebra with the operations

x ր y = x ≻ R(y), x ց y = R(x) ≻ y, x ւ y = R(x) ≺ y, x տ y = x ≺ R(y).

Theorem 4.2. Assume hypothesis of Corollary 3.2 and let (V,L≺, R≺,L≻, R≻, β) be a bimodule and T be an O-operator of (A, ≺, ≻, α) associated to (V,L≺, R≺,L≻, R≻, β). Then (V,L≻ − R≺, R≻ − L≺, β) is a bimodule of the Hom-pre-Malcev algebra (A, ·, α), and T is an O-operator of (A, ·, α) with respect to (V,L≻−R≺, R≻−L≺, β). Moreover, if (V, ◮, ◭, β) is the Hom-M-dendriform algebra associated to the Hom-pre-Malcev algebra (A, ·, α) on the bimodule (V,L≻ − R≺, R≻ − L≺, β), then

a ◮ b = a ր b − b ւ a, a ◭ b = a ց b − b տ a.

Proof. By Proposition 4.5, A ⋉ V is a Hom-pre-alternative algebra. Consider its associated Hom-pre-Malcev algebra (A ⊕ V, •, α + β),

(x + a) • (y + b) = (x + a) ≫ (y + b) − (y + b) ≪ (x + a)

= x ≻ y + L≻(x)b + R≻(y)a − y ≺ x − L≺(y)a − R≺(x)b

= x · y + (L≻ − R≺)(x)b + (R≻ − L≺)(y)a.

By Proposition 3.6, (V,L≻ − R≺, R≻ − L≺, β) is a representation of Hom-pre-Malcev algebra (A, ·, α). Moreover, T is an O-operator of (A, ·, α) with respect to (V,L≻ − R≺, R≻ − L≺, β). In fact,

T (a) · T (b)=T (a) ≻ T (b) − T (b) ≺ T (a)

=T L≻(T (a))b + R≻(T (b))a − T L≺(T (b))a + R≺(T (a))b =T (L≻ − R≺)(T (a))b + (R≻ − L≺)(T (b))a . Moreover, by (4.12) and (4.24),

a ◮ b = (R≻ − L≺)(T (b))a = R≻(T (b))a − L≺(T (b))a = a ր b − b ւ a,

a ◭ b = (L≻ − R≺)(T (a))b = L≻(T (a))b − R≺(T (a))b = a ց b − b տ a.

28 Corollary 4.4. Assume hypothesis of Corollary 3.2, and let R be a Rota-Baxter operator of (A, ≺, ≻, α) and (A, ◮, ◭, α) be the Hom-M-dendriform algebra associated to the Hom-pre-Malcev algebra (A, ·, α) given in Corollary 4.1. Then, the operations

x ◮ y = x ր y − y ւ x, x ◭ y = x ց y − y տ x, deﬁne a Hom-M-dendriform structure in A with respect the twisting map α.

Combining it with diagram (3.15), yields the following detailed commutative diagram:

◮=ր−ւ Hom-alt quadri-alg (A, տ, ւ, ր, ց, α) // Hom-M-dendri alg (A, ◮, ◭, α) ◭=ց−տ ≻=ր + ց O O R-B · =◮ + ◭ R-B ≺=տ + ւ · =≺−≻ Hom-pre-alt alg (A, ≺, ≻, α) // Hom-pre-Malcev alg (A, ·, α) O O ⋆ =≺ + ≻ R-B Commutator R-B Commutator Hom-alt alg (A, ⋆, α) // Hom-Malcev alg (A, [−, −], α)

References

[1] Aguiar, M., Loday, J.-L.: Quadri-algebras, J. Pure Applied Algebra, 191, 205- 221 (2004) [2] Ammar, F., Ejbehi, Z., Makhlouf, A.: Cohomology and deformations of Hom- algebras, J. Lie Theory 21(4), 813-836 (2011) [3] Attan, S, Laraiedh, I: Construtions and bimodules of BiHom-alternative and BiHom-Jordan algebras, arXiv:2008.07020 [math.RA] (2020) [4] Bai, C. M.: A uniﬁed algebraic approach to classical Yang-Baxter equation, J. Phy. A: Math. Theor. 40, 11073-11082 (2007) [5] Bakayoko, I.: Laplacian of Hom-Lie quasi-bialgebras, International Journal of Algebra, 8(15), 713-727 (2014) [6] Bakayoko, I.: L-modules, L-comodules and Hom-Lie quasi-bialgebras, African Diaspora Journal of Mathematics, 17, 49-64 (2014) [7] Bakayoko, I., Banagoura, M.: Bimodules and Rota-Baxter Relations. J. Appl. Mech. Eng. 4(5) (2015) [8] Bakayoko, I., Silvestrov, S.: Multiplicative n-Hom-Lie color algebras, In: Silve- strov, S., Malyarenko, A., Ran˘cić, M. (Eds.), Algebraic Structures and Applica- tions, Springer Proceedings in Mathematics and Statistics 317, Ch. 7, 159-187, Springer (2020). (arXiv:1912.10216[math.QA] (2019))

29 [9] Bakayoko, I., Silvestrov, S.: Hom-left-symmetric color dialgebras, Hom-tridendriform color algebras and Yau’s twisting generalizations, Afrika Matematika (accepted in January 2020), arXiv:1912.01441[math.RA] (2019) [10] Benayadi, S., Makhlouf, A.: Hom-Lie algebras with symmetric invariant nonde- generate bilinear forms, J. Geom. Phys. 76, 38-60 (2014) [11] Ben Abdeljelil, A., Elhamdadi, M., Kaygorodov, I., Makhlouf, A.: Gener- alized Derivations of n-BiHom-Lie algebras, In: Silvestrov, S., Malyarenko, A., Ran˘cić, M. (Eds.), Algebraic Structures and Applications, Springer Pro- ceedings in Mathematics and Statistics 317, Ch. 4, 81-97, Springer (2020). (arXiv:1901.09750[math.RA] (2019)) [12] Caenepeel, S., Goyvaerts, I.: Monoidal Hom-Hopf Algebras, Comm. Algebra 39(6), 2216-2240 (2011) [13] Chtioui, T., Mabrouk, S., Makhlouf, A.: BiHom-alternative, BiHom-Malcev and BiHom-Jordan algebras." Rocky Mountain J. Math. 50(1), 69-90 (2020) [14] Chtioui, T., Mabrouk, S., Makhlouf, A.: BiHom-pre-alternative algebras and BiHom-alternative quadri-algebras, Bull. Math. Soc. Sci. Math. Roumanie. 63 (111)(1), 3–21 (2020) [15] Dassoundo, M. L., Silvestrov, S.: Nearly associative and nearly Hom-associative algebras and bialgebras, arXiv:2101.12377 [math.RA] (2021) [16] Graziani, G., Makhlouf, A., Menini, C., Panaite, F.: BiHom-Associative Alge- bras, BiHom-Lie Algebras and BiHom-Bialgebras, SIGMA 11(086), 34 pp (2015) [17] Gürsey, F., Tze, C.-H.: On the role of division, Jordan and related algebras in particle physics, World Scientiﬁc (1996) [18] Hartwig, J. T., Larsson, D., Silvestrov, S. D.: Deformations of Lie algebras using σ-derivations, J. Algebra, 295, 314-361 (2006) (Preprint in Mathematical Sciences 2003:32, LUTFMA-5036-2003, Centre for Mathematical Sciences, De- partment of Mathematics, Lund Institute of Technology, 52 pp. (2003)) [19] Hassanzadeh, M., Shapiro, I., Sütlü, S.: Cyclic homology for Hom-associative algebras, J. Geom. Phys. 98, 40-56 (2015) [20] Hounkonnou, M. N., Dassoundo, M. L.: Center-symmetric Algebras and Bialge- bras: Relevant Properties and Consequences. In: Kielanowski P., Ali S., Bieli- avsky P., Odzijewicz A., Schlichenmaier M., Voronov T. (eds) Geometric Meth- ods in Physics. Trends in Mathematics. 2016, pp. 281-293. Birkhäuser, Cham (2016) [21] Hounkonnou, M. N., Houndedji, G. D., Silvestrov, S.: Double constructions of biHom-Frobenius algebras, arXiv:2008.06645 [math.QA] (2020)

30 [22] Hounkonnou, M. N., Dassoundo, M. L.: Hom-center-symmetric algebras and bialgebras. arXiv:1801.06539. [23] Kitouni, A., Makhlouf, A., Silvestrov, S.: On n-ary generalization of BiHom- Lie algebras and BiHom-associative algebras, In: Silvestrov, S., Malyarenko, A., RancicRan˘cić, M. (Eds.), Algebraic Structures and Applications, Springer Pro- ceedings in Mathematics and Statistics 317, Springer, Ch. 5, 99-126 (2020) [24] Laraiedh, I: Bimodules and matched pairs of noncommutative BiHom-(pre)- Poisson algebras, arXiv:2102.11364 [math.RA] (2021) [25] Kupershmidt, B. A.; What a Classical r-Matrix Really Is, J. Nonlin. Math. Phys., 6(4), 448-488 (1999) [26] Kerdman, F. S.: Analytic Moufang loops in the large, Alg. Logic 18, 325-347 (1980) [27] Kuzmin, E. N.: The connection between Malcev algebras and analytic Moufang loops, Alg. Logic, 10, 1-14 (1971) [28] Kuzmin, E. N.: Malcev algebras and their representations, Alg. Logic, 7, 233-244 (1968) [29] Larsson, D., Silvestrov, S. D.: Quasi-Hom-Lie algebras, central extensions and 2- cocycle-like identities, J. Algebra 288, 321-344 (2005) (Preprints in Mathematical Sciences 2004:3, LUTFMA-5038-2004, Centre for Mathematical Sciences, Depart- ment of Mathematics, Lund Institute of Technology, Lund University (2004)) [30] Larsson, D., Silvestrov, S. D.: Quasi-Lie algebras. In "Noncommutative Geome- try and Representation Theory in Mathematical Physics". Contemp. Math., 391, Amer. Math. Soc., Providence, RI, 241-248 (2005) (Preprints in Mathematical Sciences 2004:30, LUTFMA-5049-2004, Centre for Mathematical Sciences, De- partment of Mathematics, Lund Institute of Technology, Lund University (2004)) [31] Larsson, D., Silvestrov, S. D.: Graded quasi-Lie agebras, Czechoslovak J. Phys. 55, 1473-1478 (2005)

[32] Larsson, D., Silvestrov, S. D.: Quasi-deformations of sl2(F) using twisted derivations, Comm. Algebra, 35, 4303-4318 (2007) (Preprint in Mathematical Sciences 2004:26, LUTFMA-5047-2004, Centre for Mathematical Sciences, Lund Institute of Technology, Lund University (2004). arXiv:math/0506172 [math.RA] (2005)) [33] Larsson, D., Sigurdsson, G., Silvestrov, S. D.: Quasi-Lie deformations on the algebra F[t]/(tN ), J. Gen. Lie Theory Appl. 2(3), 201-205 (2008) [34] Larsson, D., Silvestrov, S. D.: On generalized N-complexes comming from twisted derivations, In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (Eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, Springer-Verlag, Ch. 7, 81-88 (2009)

31 [35] Loday, J-L., Dialgebras: In: Loday, J.-L., Frabetti, A., Chapoton, F., Goichot, F. (Eds), Dialgebras and Related Operads, Lecture Notes in Math. 1763, Springer, Berlin, 7-66 (2001). (Prépublication de l’Inst de Recherche Math. Avancée (Stras- bourg), 14, 61 pp (1999)) [36] Loday, J.-L., Ronco, M.: Trialgebras and families of polytopes. Contemp. Math. 346, 369-398 (2004) [37] Ma, T., Makhlouf, A., Silvestrov, S.: Curved O-operator systems, 17pp, arXiv:1710.05232 [math.RA] (2017) [38] Ma, T., Makhlouf, A., Silvestrov, S.: Rota-Baxter bisystems and covariant bialgebras, 30 pp, arXiv:1710.05161[math.RA] (2017) [39] Ma, T., Makhlouf, A., Silvestrov, S.: Rota-Baxter Cosystems and Coquasitrian- gular Mixed Bialgebras, J. Algebra Appl. Accepted 2019. (Puiblished ﬁrst online 2020: doi:10.1142/S021949882150064X) [40] Ma, T., Zheng, H.: Some results on Rota–Baxter monoidal Hom-algebras, Re- sults Math. 72 (1-2), 145-170 (2017). [41] Madariaga, S.: Splitting of operations for alternative and Malcev structures, Communications in Algebra, 45(1), 183-197 (2014) [42] Mabrouk, S., Ncib, O., Silvestrov, S.: Generalized Derivations and Rota-Baxter Operators of n-ary Hom-Nambu Superalgebras, Adv. Appl. Cliﬀord Algebras, 31, 32, (2021). (arXiv:2003.01080[math.QA]) [43] Makhlouf, A.: Hom-dendriform algebras and Rota–Baxter Hom-algebras, In: Bai, C., Guo, L., Loday, J.-L. (eds.), Nankai Ser. Pure Appl. Math. Theoret. Phys., 9, World Sci. Publ. 147–171 (2012) [44] Makhlouf, A.: Hom-alternative algebras and Hom-Jordan algebras, Int. Elect. Journ. of Alg., 8, 177-190 (2010). (arXiv:0909.0326 (2009)) [45] Makhlouf, A.: Paradigm of nonassociative Hom-algebras and Hom-superalgebras, Proceedings of Jordan Structures in Algebra and Analysis Meeting, 145-177 (2010). (arXiv:1001.4240v1) [46] Malcev, A. I.: Analytic loops, Mat. Sb. 36(3), 569-576 (1955) [47] Makhlouf, A., Silvestrov, S. D.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2(2), 51–64 (2008) (Preprints in Mathematical Sciences 2006:10, LUTFMA- 5074-2006, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, Lund University (2006)) [48] Makhlouf, A., Silvestrov, S.: Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras, In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (Eds.), General- ized Lie Theory in Mathematics, Physics and Beyond, Springer-Verlag, Berlin,

32 Heidelberg, Ch. 17, 189-206 (2009) (Preprints in Mathematical Sciences, Lund University, Centre for Mathematical Sciences, Centrum Scientiarum Mathemati- carum (2007:25) LUTFMA-5091-2007 and in arXiv:0709.2413 [math.RA] (2007)) [49] Makhlouf, A., Silvestrov, S. D.: Hom-algebras and Hom-coalgebras, J. Algebra Appl. 9(4), 553-589 (2010) (Preprints in Mathematical Sciences, Lund Univer- sity, Centre for Mathematical Sciences, Centrum Scientiarum Mathematicarum, (2008:19) LUTFMA-5103-2008. arXiv:0811.0400 [math.RA] (2008)) [50] Makhlouf, A., Silvestrov, S.: Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras, Forum Math. 22(4), 715-739 (2010) (Preprints in Mathematical Sciences, Lund University, Centre for Mathemati- cal Sciences, Centrum Scientiarum Mathematicarum, (2007:31) LUTFMA-5095- 2007. arXiv:0712.3130v1 [math.RA] (2007)) [51] Makhlouf, A., Yau, D.: Rota-Baxter Hom-Lie admissible algebras, Comm. Alg., 23(3), 1231-1257 (2014) [52] Myung, H. C.: Malcev-admissible algebras, Progress in Math. 64, Birkhäuser, (1986) [53] Nagy, P. T.: Moufang loops and Malcev algebras, Sem. Sophus Lie, 3, 65-68 (1993) [54] Okubo, S.: Introduction to octonion and other non-associative algebras in physics, Cambridge Univ. Press, (1995) [55] Richard, L., Silvestrov, S. D.: Quasi-Lie structure of σ-derivations of C[t±1], J. Algebra 319(3), 1285-1304 (2008) (arXiv:math/0608196[math.QA] (2006). Preprints in mathematical sciences (2006:12), LUTFMA-5076-2006, Centre for Mathematical Sciences, Lund University (2006)) [56] Richard, L., Silvestrov, S. D.: A note on quasi-Lie and Hom-Lie structures of σ- C ±1 ±1 derivations of [z1 ,...,zn ], In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (Eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, Springer- Verlag, Ch. 22, 257-262, (2009) [57] Saadaou, N, Silvestrov, S.: On (λ, µ, γ)-derivations of BiHom-Lie algebras, arXiv:2010.09148 [math.RA], (2020) [58] Sabinin, L. V.: Smooth quasigroups and loops, Kluwer Academic, (1999) [59] Sagle, A. A.: Malcev algebras, Trans. Amer. Math. Soc. 101, 426-458 (1961) [60] Sheng, Y.: Representations of Hom-Lie algebras, Algebr. Reprensent. Theory 15, 1081-1098 (2012) [61] Sheng, Y., Bai, C.: A new approach to Hom-Lie bialgebras, J. Algebra, 399, 232-250 (2014)

33 [62] Sigurdsson, G., Silvestrov, S.: Graded quasi-Lie algebras of Witt type, Czechoslo- vak J. Phys. 56, 1287-1291 (2006) [63] Sigurdsson, G., Silvestrov, S.: Lie color and Hom-Lie algebras of Witt type and their central extensions, In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (Eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, Springer- Verlag, Berlin, Heidelberg, Ch. 21, 247-255 (2009) [64] Silvestrov, S.: Paradigm of quasi-Lie and quasi-Hom-Lie algebras and quasi-deformations, In "New techniques in Hopf algebras and graded ring theory", K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels, 165-177 (2007) [65] Silvestrov, S., Zargeh, C.: HNN-extension of involutive multiplicative Hom-Lie algebras, arXiv:2101.01319 [math.RA] (2021) [66] Sun, Q.: On Hom-Prealternative Bialgebras, Algebr. Represent. Theor. 19, 657- 677 (2016) [67] Yau, D.: Hom-Malcev, Hom-alternative, and Hom-Jordan algebras, Int. Electron. J. Algebra, 11, 177-217 (2012) [68] Yau, D.: Module Hom-algebras, arXiv:0812.4695[math.RA] (2008) [69] Yau, D.: Enveloping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl. 2(2), 95-108 (2008). (arXiv:0709.0849 [math.RA] (2007)) [70] Yau, D.: Hom-algebras and homology, J. Lie Theory 19(2), 409-421 (2009) [71] Yau, D.: Hom-bialgebras and comodule Hom-algebras, Int. Electron. J. Algebra 8, 45-64 (2010). (arXiv:0810.4866[math.RA] (2008)) [72] Yau D.: The Hom-Yang-Baxter equation, Hom-Lie algebras and quasi-triangular bialgebras, J. Phys. A. 42, 165–202 (2006)