� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 35, Num. 1 (2020), 69 – 97 doi:10.17398/2605-5686.35.1.69 Available online April 29, 2020 Hom-Jordan and Hom-alternative bimodules S. Attan, H. Hounnon, B. Kpamegan Département de Mathématiques, Université d’Abomey-Calavi 01 BP 4521, Cotonou 01, Bénin syltane2010@yahoo.fr , hi.hounnon@fast.uac.bj , kpamegan bernadin@yahoo.fr Received March 21, 2019 and, in revised form, April 8, 2020 Presented by Consuelo Mart́ınez Accepted April 15, 2020 Abstract: In this paper, Hom-Jordan and Hom-alternative bimodules are introduced. It is shown that Jordan and alternative bimodules are twisted via endomorphisms into Hom-Jordan and Hom- alternative bimodules respectively. Some relations between Hom-associative bimodules, Hom-Jordan and Hom-alternative bimodules are given. Key words: Bimodules, alternative algebras, Jordan algebras, Hom-alternative algebras, Hom- Jordan algebras, Hom-associative algebras. AMS Subject Class. (2010): 17A30, 17B10, 17C50, 17D05. 1. Introduction Algebras where the identities defining the structure are twisted by a homo- morphism are called Hom-algebras. They have been intensively investigated in the literature recently. Hom-algebra started from Hom-Lie algebras intro- duced and discussed in [6, 10, 11, 12], motivated by quasi-deformations of Lie algebras of vector fields, in particular q-deformations of Witt and Vira- soro algebras. Hom-associative algebras were introduced in [15] while Hom- alternative and Hom-Jordan algebras are introduced in [14], [23] as twisted generalizations of alternative and Jordan algebra respectively. The reader is referred to [20] for applications of alternative algebras to projective geometry, buildings, and algebraic groups and to [4, 9, 16, 19] for discussions about the important roles of Jordan algebras in physics, especially quantum mechanics. The anti-commutator of a Hom-alternative algebra gives rise to a Hom- Jordan algebra [23]. Starting with a Hom-alternative algebra (A, ·,α), it is known that the Jordan product x∗y = 1 2 (x ·y + y ·x) gives a Hom-Jordan algebra A+ = (A,∗,α). In other words, Hom-alternative algebras are Hom-Jordan-admissible [23]. ISSN: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.35.1.69 mailto:syltane2010@yahoo.fr mailto:hi.hounnon@fast.uac.bj mailto:kpamegan_bernadin@yahoo.fr https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 70 s. attan, h. hounnon, b. kpamegan The notion of bimodule for a class of algebras defined by multilinear iden- tities has been introduced by Eilenberg [3]. If H is in the class of associative algebras or in the one of Lie algebras then this notion is the familiar one for which we are in possession of well-worked theories. The study of bimod- ule (or representation) of Jordan algebras was initiated by N. Jacobson [7]. Subsequently the alternative case was considered by Schafer [17]. Modules over an ordinary algebra has been extended to the ones of Hom- algebras in many works [2, 18, 21, 22]. The aim of this paper is to introduce Hom-alternative bimodules and Hom- Jordan bimodules and to discuss about some findings. The paper is organized as follows. In section two, we recall basic notions related to Hom-algebras and modules over Hom-associative algebras. Section three is devoted to the introduction of Hom-alternative bimodules . Proposition 3.7 shows that from a given Hom-alternative bimodule, a sequence of this kind of bimodules can be obtained. Theorem 3.8 establishes that, an alternative bimodule gives rise to a bimodule over the corresponding twisted algebra. It is also proved that a direct sum of a Hom-alternative algebra and a module over this Hom- algebra is again a Hom-alternative algebra (Theorem 3.11). In section four, we introduce Hom-Jordan modules and attest similar results as in the previous section. Furthermore, it is proved that a Hom-Jordan special left and right module, with an additional condition, has a bimodule structure over this Hom- algebra (Theorem 4.10). Finally, Proposition 4.12 shows that a bimodule over a Hom-associative algebra has a bimodule structure over its plus Hom- algebra. All vector spaces are assumed to be over a fixed ground field K of characteristic 0. 2. Preliminaries We recall some basic notions introduced in [6, 15, 21] related to Hom- algebras and while dealing of any binary operation we will use juxtaposition in order to reduce the number of braces, i.e., e.g., for “·”, xy · α(z) means (x ·y) ·α(z). Also, for the map µ : A⊗2 → A, we will write sometimes µ(a⊗b) as µ(a,b) or ab for a,b ∈ A and if V is another vector space, τ1 : A⊗V → V ⊗A (resp. τ2 : V ⊗A → A⊗V ) denote the twist isomorphism τ1(a⊗v) = v ⊗a (resp. τ2(v ⊗a) = a⊗v). Definition 2.1. A Hom-module is a pair (M,αM ) consisting of a K- module M and a linear self-map αM : M → M. A morphism f : (M,αM ) → (N,αN ) of Hom-modules is a linear map f : M → N such that f◦αM = αN◦f. hom-jordan and hom-alternative bimodules 71 Definition 2.2. ([15, 21]) A Hom-algebra is a triple (A,µA,αA) in which (A,αA) is a Hom-module, µ : A ⊗2 → A is a linear map. The Hom- algebra (A,µ,α) is said to be multiplicative if α◦µ = µ◦α⊗2 (multiplicativity). A morphism f : (A,µA,αA) → (B,µB,αB) of Hom-algebras is a morphism of the underlying Hom-modules such that f ◦µA = µB ◦f⊗2. An important class of Hom-algebras that is considered here is the one of Hom-alternative algebras. These algebras have been introduced in [14] and more studied in [23]. Definition 2.3. Let (A,µ,α) be a Hom-algebra. (i) The Hom-associator of A is the linear map asA : A ⊗3 → A defined as asA = µ ◦ (µ ⊗ α − α ⊗ µ). A multiplicative Hom-algebra (A,µ,α) is said to be Hom-associative algebra if asA = 0. (ii) A Hom-alternative algebra [14] is a multiplicative Hom-algebra (A,µ,α) that satisfies asA(x,x,y) = 0 (left Hom-alternativity) , (1) asA(x,y,y) = 0 (right Hom-alternativity) (2) for all x,y ∈ A. (iii) Let (A,µ,α) be a Hom-alternative algebra. A Hom-subalgebra of (A,µ,α) is a linear subspace H of A, which is closed for the multi- plication µ and invariant by α, that is, µ(x,y) ∈ H and α(x) ∈ H for all x,y ∈ H. If furthermore µ(a,b) ∈ H and µ(b,a) ∈ H for all (a,b) ∈ A×H, then H is called a two-sided Hom-ideal of A. Now, we prove: Proposition 2.4. Let (A,µ,α) be a Hom-alternative algebra and I be a two-sided Hom-ideal of (A,µ,α). Then (A/I,µ̄, ᾱ) is a Hom-alternative algebra where µ̄(x̄, ȳ) = µ(x,y) and ᾱ(x̄) = ¯α(x) for all x̄, ȳ ∈ A/I. Proof. First, note that the multiplicativity of µ̄ with respect to ᾱ follows from the one of µ with respect to α. Next, pick x̄, ȳ ∈ A/I. Then the left Hom-alternativity (1) in (A/I,µ̄, ᾱ) is proved as follows asA/I(x̄, x̄, ȳ) = µ̄(µ̄(x̄, x̄), ᾱ(ȳ)) − µ̄(ᾱ(x̄), µ̄(x̄, ȳ) = µ(µ(x,x)α(y)) −µ(α(x),µ(x,y)) = asA(x,x,y)) = 0̄ . Similarly, we get (2) and therefore (A/I,µ̄, ᾱ) is a Hom-alternative algebra. 72 s. attan, h. hounnon, b. kpamegan As Hom-alternative algebras, Hom-Jordan algebras are fundamental ob- jects of this paper. They appear as cousins of Hom-alternative algebras and these two Hom-algebras are related as Jordan and alternative algebras. Definition 2.5. ([23]) (i) A Hom-Jordan algebra is a multiplicative Hom-algebra (A,µ,α) such that µ◦τ = µ (commutativity of µ) and the so-called Hom-Jordan identity holds asA(µ(x,x, ),α(y),α(x)) = 0,∀ (x,y) ∈ A2 (3) where, τ : A⊗2 → A⊗2, τ(a⊗ b) = b⊗a, is the twist isomorphism. (ii) Let (A,µ,α) be a Hom-Jordan algebra. A Hom-subalgebra of (A,µ,α) is a linear subspace H of A, which is closed for the multiplication µ and invariant by α, that is, µ(x,y) ∈ H and α(x) ∈ H for all x,y ∈ H. If furthermore µ(a,b) ∈ H for all (a,b) ∈ A × H, then H is called a two-sided Hom-ideal (or simply Hom-ideal) of A [5]. Similarly as a Hom-alternative algebra case, if H is a Hom-ideal of a Hom- Jordan algebra (A,µ,α), then (A/H,µ̄, ᾱ) is a Hom-Jordan algebra where µ̄(x̄, ȳ) = µ(x,y) for all x̄, ȳ ∈ A/H and ᾱ : A/H → A/H is naturally induced by α, inherits a Hom-Jordan algebra structure, which is named quotient Hom- Jordan algebra. Remark 2.6. In [14] Makhlouf defined a Hom-Jordan algebra as a com- mutative multiplicative Hom-algebra satisfying asA(x 2,y,α(x)) = 0, which becomes the identity (3) if y is replaced by α(y). The proof of the following result can be found in [23] where the product ∗, differs from the one given here by a factor of 1 2 . Proposition 2.7. Let (A,µ,α) be a Hom-alternative algebra. Then A+ = (A,∗,α) is a Hom-Jordan algebra where x∗y = xy + yx for all x,y ∈ A. Example 2.8. From the eight-dimensional Hom-alternative algebra Oα = (O,µα,α) with basis {e0,e1,2 ,e3,e4,e5,e6,e7} [23, Example 3.19], constructed from the octonion algebra which is an eight-dimensional alternative algebra, we obtain, the Hom-Jordan algebra O+α = (O,∗ = µα+µα◦τ,α) where the non zero products are: e0 ∗e0 = 2e0, e0 ∗e1 = e1 ∗e0 = 2e5, e0 ∗e2 = e2 ∗e0 = 2e6, e0 ∗ e3 = e3 ∗ e0 = 2e7, e0 ∗ e4 = e4 ∗ e0 = 2e1, e0 ∗ e5 = e5 ∗ e0 = 2e2, e0 ∗ e6 = e6 ∗ e0 = 2e3, e0 ∗ e7 = e7 ∗ e0 = 2e4, e1 ∗ e1 = e2 ∗ e2 = e3 ∗ e3 = hom-jordan and hom-alternative bimodules 73 e4 ∗ e4 = e5 ∗ e5 = e6 ∗ e6 = e7 ∗ e7 = −2e0 and the twisting map α is given by α(e0) = e0, α(e1) = e5, α(e2) = e6, α(e3) = e7, α(e4) = e1, α(e5) = e2, α(e6) = e3, α(e7) = e4. A. Makhlouf proved that the plus algebra of any Hom-associative algebra is a Hom-Jordan algebra as defined in [14]. Here, we prove the same result for the Hom-Jordan algebra as defined in [23] (see also Definition 2.5 above). Proposition 2.9. Let (A, ·,α) be a Hom-associative algebra. Then A+ = (A,∗,α) is a Hom-Jordan algebra where x∗y = xy + yx for all x,y ∈ A. Proof. The commutativity of ∗ is obvious. We compute the Hom-Jordan identity as follows: asA+ ( x2,α(x),α(y) ) = (x2 ∗α(y)) ∗α2(x) −α(x2) ∗ (α(y) ∗α(x)) = (x2 ·α(y)) ·α2(x) + (α(y) ·x2) ·α2(x) + α2(x) · (x2 ·α(y)) + α2(x) · (α(y) ·x2) −α(x2) · (α(y) ·α(x)) −α(x2) · (α(x) ·α(y)) − (α(y) ·α(x)) ·α(x2) − (α(x) ·α(y)) ·α(x2) (by a direct computation) = (α(y) ·x2) ·α2(x) + α2(x) · (x2 ·α(y)) −α(x2) · (α(x) ·α(y)) − (α(y) ·α(x)) ·α(x2) (by the Hom-associativity) = (α(y) ·x2) ·α2(x) + α2(x) · (x2 ·α(y)) − (α(x) ·α(x)) ·α(x ·y) −α(yx) · (α(x) ·α(x)) (by the multiplicativity) = (α(y) ·x2) ·α2(x) + α2(x) · (x2 ·α(y)) −α2(x) · (α(x) · (x ·y)) − ((yx) ·α(x)) ·α2(x) (by the Hom-associativity) = 0 (by the Hom-associativity) . Then A+ = (A,∗,α) is a Hom-Jordan algebra. Examples 2.10. (i) Consider the three-dimensional Hom-associative al- gebra A = (A,µA,αA) over K with basis (e1,e2,e3) defined by µA(e1,e1) = e1, µA(e2,e2) = e2, µA(e3,e3) = e1, µA(e1,e3) = µA(e3,e1) = −e3 and αA(e1) = e1, αA(e3) = −e3 (see [24, Theorem 3.12], Hom-algebra A′ 3 3). Using the prod- uct ∗ in Proposition 2.9, the triple A+ = (A,∗,αA) is Hom-Jordan algebra where, e1 ∗e1 = 2e1, e2 ∗e2 = 2e2, e3 ∗e3 = 2e1, e1 ∗e3 = e3 ∗e1 = −2e3. 74 s. attan, h. hounnon, b. kpamegan (ii) From the three-dimensional Hom-associative algebra B = (B,µB,αB) over K with basis (e1,e2,e3) defined by µB(e1,e1) = e1, µB(e2,e2) = e1, µB(e3,e3) = e3, µB(e1,e2) = µB(e2,e1) = −e2 and αB(e1) = e1, αB(e2) = −e2 (see [24, Theorem 3.12], Hom-algebra A′ 3 5). Then the triple B+ = (B,∗, αB) is a Hom-Jordan algebra, where “∗” is the product in Proposition 2.9 and e1 ∗e1 = 2e1, e2 ∗e2 = 2e1, e3 ∗e3 = 2e3, e1 ∗e2 = e2 ∗e1 = −2e2. Let us consider the following definitions which will be used in next sections. Definition 2.11. Let (A,µ,αA) be any Hom-algebra. (i) A Hom-module (V,αV ) is called an A-bimodule if it comes equipped with a left and a right structure maps on V that is morphisms ρl : A⊗V → V , a ⊗ v 7→ a · v and ρr : V ⊗ A → V , v ⊗ a 7→ v · a of Hom-modules respectively. (ii) A morphism f : (V,αV ,ρl,ρr) → (W,αW ,ρ′l,ρ ′ r) of A-bimodules is a morphism of the underlying Hom-modules such that f ◦ρl = ρ′l ◦ (IdA ⊗f) and f ◦ρr = ρ ′ r ◦ (f ⊗ IdA) . (iii) Let (V,αV ) be an A-bimodule with structure maps ρl and ρr. Then the module Hom-associator of V is a trilinear map asA,V defined as: asA,V ◦ IdV⊗A⊗A = ρr ◦ (ρr ⊗αA) −ρl ◦ (αV ⊗µ) , asA,V ◦ IdA⊗V⊗A = ρr ◦ (ρl ⊗αA) −ρl ◦ (αA ⊗ρr) , asA,V ◦ IdA⊗A⊗V = ρl ◦ (µ⊗αV ) −ρl ◦ (αA ⊗ρl) . Remark 2.12. The module Hom-associator given above is a generalization of the one given in [2]. Now, let consider the following notion for Hom-associative algebras. Definition 2.13. Let (A,µ,αA) be a Hom-associative algebra and (M,αM ) be a Hom-module. (i) A left Hom-associative A-module structure on M consists of a morphism ρ : A⊗M → M of Hom-modules, such that ρ◦ (αA ⊗ρ) = ρ◦ (µ⊗αM ) (4) hom-jordan and hom-alternative bimodules 75 (ii) A right Hom-associative A-module structure on M consists of a mor- phism ρ : M ⊗A → M of Hom-modules, such that ρ◦ (αM ⊗µ) = ρ◦ (ρ⊗αA) (5) (iii) A Hom-associative A-bimodule structure on M consists of two structure maps ρl : A⊗M → M and ρr : M ⊗A → M such that (M,αM,ρl) is a left A-module, (M,αM,ρr) is a right A-module and that the following Hom-associativity (or operator commutativity) condition holds: ρl ◦ (αA ⊗ρr) = ρr ◦ (ρl ⊗αA) (6) Remark 2.14. Actually, left Hom-associative A-module, right Hom-asso- ciative A-module and Hom-associative A-bimodule have been already intro- duced in [21, 22] where they are called left A-module, right A-module and A-bimodule respectively. The expressions, used in Definition 2.13 for these notions, are motivated by the unification of our terminologies. 3. Hom-alternative bimodules In this section, we give the definition of Hom-alternative (bi)modules. We prove that from a given Hom-alternative bimodule, a sequence of this kind of bimodules can be constructed. It is also proved that a direct sum of a Hom- alternative algebra and a bimodule over this Hom-algebra is a Hom-alternative algebra called a split null extension of the considered Hom-algebra. First, we start by the following notion, due to [2], where it is called a module over a left (resp. right) Hom-alternative algebra. However, we call it a Hom-alternative left (resp. right) module in this paper. Definition 3.1. Let (A,µ,αA) be a Hom-alternative algebra. (i) A left Hom-alternative A-module is a Hom-module (V,αV ) with a left structure map ρl : A⊗V → V , a⊗v 7→ a ·v such that asA,V (x,y,v) = −asA,V (y,x,v) for all x,y ∈ A and v ∈ V . (ii) A right Hom-alternative A-module is a Hom-module (V,αV ) with a right structure map ρr : V ⊗A → V , v ⊗a 7→ v ·a such that asA,V (v,x,y) = −asA,V (v,y,x) for all x,y ∈ A and v ∈ V . 76 s. attan, h. hounnon, b. kpamegan Now, as a generalization of alternative bimodules [8, 17], one has: Definition 3.2. Let (A,µ,αA) be a Hom-alternative algebra. A Hom- alternative A-bimodule is a Hom-module (V,αV ) with a (left) structure map ρl : A⊗V → V , a⊗v 7→ a ·v and a (right) structure map ρr : V ⊗A → V , v ⊗a 7→ v ·a such that the following equalities hold: asA,V (a,v,b) = −asA,V (v,a,b) = asA,V (b,a,v) = −asA,V (a,b,v) (7) for all (a,b,v) ∈ A×2 ×V . Remarks 3.3. (i) The relation (7) is equivalent to asA,V (a,v,b) = −asA,V (v,a,b) = asA,V (b,a,v) = −asA,V (b,v,a) or since the field’s characteristic is 0 to asA,V (a,v,b) = −asA,V (v,a,b) = asV (b,a,v) and asA,V (a,a,v) = 0 . (ii) If αA = IdA and αV = IdV then V is the so-called alternative bimodule for the alternative algebra (A,µ) [8, 17]. Examples 3.4. Here are some examples of Hom-alternative A-bimodules. (i) Let (A,µ,αA) be a Hom-alternative algebra. Then (A,αA) is a Hom- alternative A-bimodule where the structure maps are ρl(a,b) = µ(a,b) and ρr(a,b) = µ(b,a). More generally, if B is a two-sided Hom-ideal of (A,µ,αA), then (B,αA) is a Hom-alternative A-bimodule where the structure maps are ρl(a,x) = µ(a,x) and ρr(x,b) = µ(x,b) for all x ∈ B and (a,b) ∈ A×2. (ii) If (A,µ) is an alternative algebra and M is an alternative A-bimodule [8] in the usual sense, then (M,IdM ) is a Hom-alternative A-bimodule where A = (A,µ,IdA) is a Hom-alternative algebra. (iii) If f : (A,µA,αA) → (B,µB,αB) is a surjective morphism of Hom- alternative algebras, then (B,αB) becomes a Hom-alternative A-bimodule via f, i.e, the structure maps are defined as ρl : (a,b) 7→ µB(f(a),b) and ρr : (b,a) 7→ µB(b,f(a)) for all (a,b) ∈ A × B. Indeed one can remark that asA,B ◦ (IdA ⊗f ⊗ IdA) = f ◦asA. In order to give another example of Hom-alternative bimodules , let us consider the following hom-jordan and hom-alternative bimodules 77 Definition 3.5. An abelian extension of Hom-alternative algebras is a short exact sequence of Hom-alternative algebras 0 → (V,αV ) i−→ (A,µA,αA) π−−→ (B,µB,αB) → 0 where (V,αV ) is a trivial Hom-alternative algebra, i and π are morphisms of Hom-algebras. Furthermore, if there exists a morphism s : (B,µB,αB) → (A,µA,αA) such that π◦s = idB then the abelian extension is said to be split and s is called a section of π. Example 3.6. Given an abelian extension as in the previous definition, the Hom-module (V,αV ) inherits a structure of a Hom-alternative B-bimodule and the actions of the Hom-algebra (B,µB,αB) on V are as follows. For any x ∈ B, there exist x̃ ∈ A such that x = π(x̃). Let x acts on v ∈ V by x ·v := µA(x̃, i(v)) and v ·x := µA(i(v), x̃). These are well-defined, as another lift x̃′ of x is written x̃′ = x̃+v′ for some v′ ∈ V and thus x·v = µA(x̃, i(v)) = µA(x̃′, i(v)) and v · x = µA(i(v), x̃) = µA(i(v), x̃′) because V is trivial. The actions property follow from the Hom-alternativity identity. In case these actions of B on V are trivial, one speaks of a central extension. The following result describes a sequence of Hom-alternative bimodules by twisting the structure maps of a given bimodule over this Hom-algebra. Proposition 3.7. Let (A,µ,αA) be a Hom-alternative algebra and (V,αV ) be a Hom-alternative A-bimodule with the structure maps ρl and ρr. Then the maps ρ (n) l = ρl ◦ (α n A ⊗ IdV ) ρ(n)r = ρr ◦ (IdV ⊗α n A) give the Hom-module (V,αV ) the structure of a Hom-alternative A-bimodule that we denote by V (n) Proof. It is clear that ρ (n) l and ρ (n) r are structure maps on V (n). Next, observe that for all x,y ∈ A and v ∈ V , asA,V (n) (x,v,y) = ρ (n) r (ρ (n) l (x,v),αA(y)) −ρ (n) l (αA(x),ρ (n) r (v,y)) = ρr(ρl(α n A(x),v),α n+1 A (y)) −ρl(α n+1 A (x),ρr(v,α n A(y)) = asA,V (α n A(x),v,α n A(y)) 78 s. attan, h. hounnon, b. kpamegan and similarly asA,V (n) (v,x,y) = asA,V (v,α n A(x),α n A(y)) , asA,V (n) (y,x,v) = asA,V (α n A(y),α n A(x),v) , asA,V (n) (x,y,v) = asA,V (α n A(x),α n A(y),v) . Therefore, equalities of (7) in V (n) derive from the one in V . We know that alternative algebras can be deformed into Hom-alternative algebras via an endomorphism. The following result shows that alternative bimodules can be deformed into Hom-alternative bimodules via an endomor- phism. This provides a large class of examples of Hom-alternative bimodules. Theorem 3.8. Let (A,µ) be an alternative algebra, V be an alternative A-bimodule with the structure maps ρl and ρr, αA be an endomorphism of the alternative algebra A and αV be a linear self-map of V such that αV ◦ρl = ρl ◦ (αA ⊗αV ) and αV ◦ρr = ρr ◦ (αV ⊗αA). Write AαA for the Hom-alternative algebra (A,µαA,αA) and VαV for the Hom-module (V,αV ). Then the maps ρ̃l = αV ◦ρl and ρ̃r = αV ◦ρr give the Hom-module VαV the structure of a Hom-alternative AαA-bimodule. Proof. Trivially, ρ̃l and ρ̃r are structure maps on VαV . The proof of (7) for VαV follows directly by the fact that asA,VαV = α 2 V ◦ asA,V and the relation (7) in V . Corollary 3.9. Let (A,µ) be an alternative algebra, V be an alternative A-bimodule with the structure maps ρl and ρr, αA an endomorphism of the alternative algebra A and αV be a linear self-map of V such that αV ◦ ρl = ρl ◦ (αA ⊗αV ) and αV ◦ρr = ρr ◦ (αV ⊗αA). Write AαA for the Hom-alternative algebra (A,µαA,αA) and VαV for the Hom-module (V,αV ). Then the maps ρ̃l (n) = ρl ◦ ( αn+1A ⊗αV ) and ρ̃r (n) = ρr ◦ ( αV ⊗αn+1A ) give the Hom-module Vα the structure of a Hom-alternative AαA-bimodule for each n ∈ N. hom-jordan and hom-alternative bimodules 79 Lemma 3.10. Let (A,µ,αA) be a Hom-alternative algebra and (V,αV ) be a Hom-alternative A-bimodule with the structure maps ρl and ρr. Then the following relation asA,V (v,a,a) = 0 (8) holds for all a ∈ A and v ∈ V . Proof. Using (7), for all (a,b) ∈ A×2 and v ∈ V we have −asA,V (v,a,b) = asA,V (a,v,b) and asA,V (v,b,a) = −asA,V (a,b,v). Moreover again from (7), we get asA,V (a,v,b) = −asA,V (a,b,v) and then −asA,V (v,a,b) = asA,V (v,b,a). It follows that asA,V (v,a,a) = 0 since the field K is of characteristic 0. The following result shows that a direct sum of a Hom-alternative algebra and a bimodule over this Hom-algebra, is still a Hom-alternative, called the split null extension determined by the given bimodule. Theorem 3.11. Let (A,µ,αA) be a Hom-alternative algebra and (V,αV ) be a Hom-alternative A-bimodule with the structure maps ρl and ρr. Defining on A⊕V the bilinear map µ̃ : (A⊕V )⊗2 → A⊕V , µ̃(a + m,b + n) := ab + a · n + m ·b and the linear map α̃ : A⊕V → A⊕V , α̃(a + m) := αA(a) + αV (m), then E = (A⊕V,µ̃, α̃) is a Hom-alternative algebra. Proof. The multiplicativity of α̃ with respect to µ̃ follows from the one of α with respect to µ and the fact that ρl and ρr are morphisms of Hom-modules. Next asE(a + m,a + m,b + n) = µ̃(µ̃(a + m,a + m), α̃(b + n)) − µ̃(α̃(a + m), µ̃(a + m,b + n)) = µ̃(a2 + a ·m + m ·a,αA(b) + αV (n)) − µ̃(αA(a) + αV (m),ab + a ·n + m · b) = a2αA(b) + a 2 ·αV (n) + (a ·m) ·αA(b) + (m ·a) ·αA(b) −αA(a)(ab) −αA(a) · (a ·n) −αA(a) · (m · b) −αV (m) · (ab) = asA(a,a,b)︸ ︷︷ ︸ 0 + asV (a,a,n)︸ ︷︷ ︸ 0 + asA,V (a,m,b) + asA,V (m,a,b)︸ ︷︷ ︸ 0 (by (1), Remarks 3.3 and (7)) = 0 . 80 s. attan, h. hounnon, b. kpamegan Similarly, we compute asE(a + m,b + n,b + n) = µ̃(µ̃(a + m,b + n), α̃(b + n)) − µ̃(α̃(a + m), µ̃(b + n,b + n)) = µ̃(ab + a ·n + m · b,αA(b) + αV (n)) − µ̃(αA(a) + αV (m),b2 + b ·n + b · b) = (ab)αA(b) + (ab) ·αV (m) + (a ·n) ·αA(b) + (m · b) ·αA(b) −αA(a)(b2) −αA(a) · (b ·n) −αA(a) · (n · b) −αV (m) · b2 = asA(a,b,b)︸ ︷︷ ︸ 0 + asA,V (a,b,n) + asA,V (a,n,b)︸ ︷︷ ︸ 0 + asA,V (m,b,b)︸ ︷︷ ︸ 0 (by (2), (7) and (8)) = 0 . We then conclude that (A⊕V,µ̃, α̃) is a Hom-alternative algebra. Remark 3.12. Consider the split null extension A⊕V determined by the Hom-alternative bimodule (V,αV ) of the Hom-alternative algebra (A,µ,αA) in the previous theorem. Write elements a + v of A ⊕ V as (a,v). Then, there is an injective homomorphism of Hom-modules i : V → A⊕V given by i(v) = (0,v) and a surjective homomorphism of Hom-modules π : A⊕V → A given by π(a,v) = a. Moreover i(V ) is a two-sided Hom-ideal of A⊕V such that A ⊕ V/i(V ) ∼= A. On the other hand, there is a morphism of Hom- algebras σ : A → A⊕V given by σ(a) = (a, 0) which is clearly a section of π. Hence, we obtain the abelian split exact sequence of Hom-alternative algebras and (V,αV ) is a Hom-alternative A-bimodule via π. 4. Hom-Jordan bimodules In this section, we study Hom-Jordan bimodules. It is observed that sim- ilar results for Hom-alternative bimodules hold for Hom-Jordan bimodules. Some of them require an additional condition. Furthermore, relations be- tween Hom-associative bimodules and Hom-Jordan bimodules are given on the one hand, and on the other hand, relations between left (resp. right) Hom-alternative modules and left(resp. right) special Hom-Jordan modules are proved. First, we have: Definition 4.1. Let (A,µ,αA) be a Hom-Jordan algebra. hom-jordan and hom-alternative bimodules 81 (i) A right Hom-Jordan A-module is a Hom-module (V,αV ) with a right structure map ρr : V ⊗ A → V , v ⊗ a 7→ v · a such that the following conditions hold: αV (v ·a) ·αA(bc) + αV (v · b) ·αA(ca) + αV (v · c) ·αA(ab) = (αV (v) · bc) ·α2A(a) + (αV (v) · ca) ·α 2 A(b) + (αV (v) ·ab) ·α2A(c) , (9) αV (v ·a) ·αA(bc) + αV (v · b) ·αA(ca) + αV (v · c) ·αA(ab) = ((v ·a) ·αA(b)) ·α2A(c) + ((v · c) ·αA(b)) ·α 2 A(a) + α2V (v) · ((ac)αA(b)) (10) for all a,b,c ∈ A and v ∈ V . (ii) A left Hom-Jordan A-module is a Hom-module (V,αV ) with a left struc- ture map ρl : A⊗V → V , a⊗v 7→ a·v such that the following conditions hold: αA(bc) ·αV (a ·v) + αA(ca) ·αV (b ·v) + αA(ab) ·αV (c ·v) = α2A(a) · (bc ·αV (v)) + α 2 A(b) · (ca ·αV (v)) + α2A(c) · (ab ·αV (v)) , (11) αA(bc) ·αV (a ·v) + αA(ca) ·αV (b ·v) + αA(ab) ·αV (c ·v) = α2A(c) · (αA(b) · (a ·v)) + α 2 A(a) · (αA(b) · (c ·v)) + ((ac)αA(b)) ·α2V (v) (12) for all a,b,c ∈ A and v ∈ V . The following result allows to introduce the notion of right special Hom- Jordan modules. Theorem 4.2. Let (A,µ,αA) be a Hom-Jordan algebra, (V,αV ) be a Hom-module and ρr : V ⊗ A → V , a ⊗ v 7→ v · a, be a bilinear map satisfying αV ◦ρr = ρr ◦ (αV ⊗αA) (13) and αV (v) · (ab) = (v ·a) ·αA(b) + (v · b) ·αA(a) (14) for all (a,b) ∈ A×2 and v ∈ V . Then (V,α,ρr) is a right Hom-Jordan A- module called a right special Hom-Jordan A-module. 82 s. attan, h. hounnon, b. kpamegan Proof. It suffices to prove (9) and (10). For all (a,b) ∈ A×2 and v ∈ V , we have: αV (v ·a) ·αA(bc) + αV (v · b) ·αA(ca) + αV (v · c) ·αA(ab) = αV (v ·a) ·αA(b)αA(c) + αV (v · b) ·αA(c)αA(a) + αV (v · c) ·αA(a)αA(b) (multiplicativity) = ((v ·a) ·αA(b)) ·α2(c) + ((v ·a) ·αA(c)) ·α2(b) + ((v · b) ·αA(c)) ·α2(a) + ((v · b) ·αA(a)) ·α2(c) + ((v · c) ·αA(a)) ·α2(b) + ((v · c) ·αA(b)) ·α2(a) (by (14)) = [αV (v) ·ab− (v · b)αA(a)] ·α2(c) + ((v ·a) ·αA(c)) ·α2(b) + [αV (v) · bc− (v · c)αA(b)] ·α2(a) + ((v · b) ·αA(a)) ·α2(c) + [αV (v) · ca− (v ·a) ·αA(c)] ·α2(b) + ((v · c) ·αA(b)) ·α2(a) (again by (14)) = (αV (v) · bc) ·α2A(a) + (αV (v) · ca) ·α 2 A(b) + (αV (v) ·ab) ·α 2 A(c) and thus, we get (9). Finally, (10) is proved as follows: αV (v ·a) ·αA(bc) + αV (v · b) ·αA(ca) + αV (v · c) ·αA(ab) = αV (v ·a) ·αA(b)αA(c) + αV (v · b) ·αA(c)αA(a) + αV (v · c) ·αA(a)αA(b) (multiplicativity) = ((v ·a) ·αA(b)) ·α2A(c) + ((v ·a) ·αA(c)) ·α 2 A(b) + ((v · b) ·αA(c)) ·α2A(a) + ((v · b) ·αA(a)) ·α 2 A(c) + ((v · c) ·αA(a)) ·α2A(b) + ((v · c) ·αA(b)) ·α 2 A(a) (by ((14)) = ((v ·a) ·αA(b)) ·α2A(c) + [αV (v) ·ac− ((v · c) ·αA(a)] ·α 2 A(b) + ((v · b) ·αA(c)) ·α2A(a) + ((v · b) ·αA(a)) ·α 2 A(c) + ((v · c) ·αA(a)) ·α2A(b) + ((v · c) ·αA(b)) ·α 2 A(a) (again by (14)) = ((v ·a) ·αA(b)) ·α2A(c) + α 2 V (v) · ((ac)αA(b)) − (αV (v) ·αA(b)) ·αA(ac) + ((v · b) ·αA(c)) ·α2A(a) + ((v · b) ·αA(a)) ·α2A(c) + ((v · c) ·αA(b)) ·α 2 A(a) (again by (14)) hom-jordan and hom-alternative bimodules 83 = ((v ·a) ·αA(b)) ·α2A(c) + α 2 V (v) · ((ac)αA(b)) − (αV (v · b) ·αA(ac) + ((v · b) ·αA(c)) ·α2A(a) + ((v · b) ·αA(a)) ·α 2 A(c) + ((v · c) ·αA(b)) ·α2A(a) (by (13)) = ((v ·a) ·αA(b)) ·α2A(c) + α 2 V (v) · ((ac)αA(b)) − ((v · b) ·αA(a)) ·α 2 A(c) − ((v · b) ·αA(c)) ·α2A(a) + ((v · b) ·αA(c)) ·α 2 A(a) + ((v · b) ·αA(a)) ·α2A(c) + ((v · c) ·αA(b)) ·α 2 A(a) (by (14)) = ((v ·a) ·αA(b)) ·α2A(c) + ((v · c) ·αA(b)) ·α 2 A(a) + α 2 V (v) · ((ac)αA(b)) which is (10). Similarly, the following result can be proved. Theorem 4.3. Let (A,µ,αA) be a Hom-Jordan algebra, (V,αV ) be a Hom-module and ρl : A ⊗ V → V , v ⊗ a 7→ a · v, be a bilinear map satisfying αV ◦ρl = ρl ◦ (αA ⊗αV ) and (ab) ·αV (v) = αA(a) · (b ·v) + αA(b) · (a ·v) (15) for all (a,b) ∈ A×2 and v ∈ V . Then (V,α,ρl) is a left Hom-Jordan A-module called a left special Hom-Jordan A-module. It is well known that the plus algebra of any Hom-alternative algebra is a Hom-Jordan algebra. The next result shows that any left (resp. right) Hom-alternative module a is also a left (resp. right) module over its plus Hom-algebra. Proposition 4.4. Let (A,µ,αA) be a Hom-alternative algebra and (V,αV ) be a Hom-module. (i) If (V,αV ) is a right Hom-alternative A-module with the structure map ρr then (V,αV ) is a right special Hom-Jordan A +-module with the same structure map ρr. (ii) If (V,αV ) is a left Hom-alternative A-module with the structure map ρl then (V,αV ) is a left special Hom-Jordan A +-module with the same structure map ρl. 84 s. attan, h. hounnon, b. kpamegan Proof. It suffices to prove (14) and (15). (i) If (V,αV ) is a right Hom-alternative A-module with the structure map ρr, then for all (x,y,v) ∈ A × A × V , asA,V (v,x,y) = −asA,V (v,y,x) by (8), i.e., αV (v) · (xy) + αV (v) · (yx) = (v · x) · αA(y) + (v · y) · αA(x). Thus αV (v) · (x ∗ y) = αV (v) · (xy) + αV (v) · (yx) = (v · x) · αA(y) + (v · y) · αA(x). Therefore (V,αV ) is a right special Hom-Jordan A+-module by Theorem 4.2. (ii) If (V,αV ) is a left Hom-alternative A-module with the structure map ρl, then for all (x,y,v) ∈ A × A × V , asA,V (x,y,v) = −asA,V (y,x,v) by Remarks 3.3 and then (xy) ·αV (v) + (yx) ·αV (v) = αA(x) · (y · v) + αA(y) · (x ·v). Thus (x∗y) ·αV (v) = (xy) ·αV (v) + (yx) ·αV (v) = αA(x) · (y ·v) + αA(y) · (x ·v). Therefore (V,αV ) is a left special Hom-Jordan A+-module by Theorem 4.3. Now, we give the definition of a Hom-Jordan bimodule. Definition 4.5. Let (A,µ,αA) be a Hom-Jordan algebra. A Hom-Jordan A-bimodule is a Hom-module (V,αV ) with a left structure map ρl : A⊗V → V , a⊗v 7→ a ·v and a right structure map ρr : V ⊗A → V , v ⊗a 7→ v ·a, such that the following conditions hold: ρr ◦ τ1 = ρl , (16) αV (v ·a) ·αA(bc) + αV (v · b) ·αA(ca) + αV (v · c) ·αA(ab) = (αV (v) · bc) ·α2A(a) + (αV (v) · ca) ·α 2 A(b) + (αV (v) ·ab) ·α2A(c) , (17) αV (v ·a) ·αA(bc) + αV (v · b) ·αA(ca) + αV (v · c) ·αA(ab) = ((v ·a) ·αA(b)) ·α2A(c) + ((v · c) ·αA(b)) ·α 2 A(a) + ((ac)αA(b)) ·α2V (v) , (18) for all a,b,c ∈ A and v ∈ V . In term of the module Hom-associator, using the relation (16) and the fact that the structure maps are morphisms, the relations (17) and (18) are respectively (a,b,c) asA,V (αA(a),αV (v),bc) = 0 , (19) asA,V (v ·a,αA(b),αA(c)) + asA,V (v · c,αA(b),αA(a)) + asA,V (ac,αA(b),αV (v)) = 0 . (20) hom-jordan and hom-alternative bimodules 85 Remarks 4.6. (i) One can note that (17) and (18) are the same identities as (9) and (10) respectively. (ii) Since ρr ◦ τ1 = ρl, nothing is lost in dropping one of the compo- sitions. Thus the term Hom-Jordan module can be used for Hom-Jordan bimodule. (iii) Since the field is of characteristic 0, the identity (19) implies asA,V (αA(a),αV (v),a 2) = 0 . (iv) If αA = IdA and αV = IdV then V is reduced to the so-called Jordan module of the Jordan algebra (A,µ) [7, 8]. Examples 4.7. Here are some examples of Hom-Jordan bimodules. (i) Let (A,µ,αA) be a Hom-Jordan algebra. Then (A,αA) is a Hom- Jordan A-bimodule where the structure maps are ρl = ρr = µ. More generally, if B is a Hom-ideal of (A,µ,αA), then (B,αA) is a Hom-Jordan A-bimodule where the structure maps are ρl(a,x) = µ(a,x) = µ(x,a) = ρr(x,a) for all (a,x) ∈ A×B. (ii) If (A,µ) is a Jordan algebra and M is a Jordan A-bimodule [8] in the usual sense then (M,IdM ) is a Hom-Jordan A-bimodule where A = (A,µ,IdA) is a Hom-Jordan algebra. (iii) If f : (A,µA,αA) → (B,µB,αB) is a surjective morphism of Hom- Jordan algebras, then (B,αB) becomes a Hom-Jordan A-bimodule via f, i.e, the structure maps are defined by ρl : (a,b) 7→ µB(b,f(a)) and ρr : (b,a) 7→ µB(f(a),b) for all (a,b) ∈ A×B. As in the case of Hom-alternative algebras, in order to give another exam- ple of Hom-Jordan bimodules, let us consider the following Definition 4.8. An abelian extension of Hom-Jordan algebras is a short exact sequence of Hom-Jordan algebras 0 → (V,αV ) i−→ (A,µA,αA) π−−→ (B,µB,αB) → 0 where (V,αV ) is a trivial Hom-Jordan algebra, i and π are morphisms of Hom-algebras. Furthermore, if there exists a morphism s : (B,µB,αB) → (A,µA,αA) such that π◦s = idB then the abelian extension is said to be split and s is called a section of π. Example 4.9. Given an abelian extension as in the previous definition, the Hom-module (V,αV ) inherits a structure of a Hom-Jordan B-bimodule 86 s. attan, h. hounnon, b. kpamegan and the actions of the Hom-algebra (B,µB,αB) on V are as follows. For any x ∈ B, there exist x̃ ∈ A such that x = π(x̃). Let x acts on v ∈ V by x ·v := µA(x̃, i(v)) and v ·x := µA(i(v), x̃). These are well-defined, as another lift x̃′ of x is written x̃′ = x̃+v′ for some v′ ∈ V and thus x·v = µA(x̃, i(v)) = µA(x̃′, i(v)) and v · x = µA(i(v), x̃) = µA(i(v), x̃′) because V is trivial. The actions property follow from the Hom-Jordan identity. In case these actions of B on V are trivial, one speaks of a central extension. The next result shows that a special left and right Hom-Jordan module has a Hom-Jordan bimodule structure under a specific condition. Theorem 4.10. Let (A,µ,αA) be a Hom-Jordan algebra and (V,αV ) be both a left and a right special Hom-Jordan A-module with the structure maps ρ1 and ρ2 respectively such that the Hom-associativity (or operator commu- tativity) condition holds ρ2 ◦ (ρ1 ⊗αA) = ρ1 ◦ (αA ⊗ρ2) . (21) Define the bilinear maps ρl : A⊗V → V and ρr : V ⊗A → V by ρl = ρ1 + ρ2 ◦ τ1 and ρr = ρ1 ◦ τ2 + ρ2 . (22) Then (V,αV ,ρl,ρr) is a Hom-Jordan A-bimodule. Proof. It is clear that ρl and ρr are structure maps and (16) holds. To prove relations (17) and (18), let put ρl(a⊗v) := a�v, i.e., a�v = a ·v + v ·a for all (a,v) ∈ A×V . We have then ρr(v ⊗a) := v �a = a ·v + v ·a for all (a,v) ∈ A×V . Therefore for all (a,b,v) ∈ A×A×V , we have αV (v �a) �αA(bc) + αV (v � b) �αA(ca) + αV (v � c) �αA(ab) = αV (v ·a) ·αA(bc) + αV (a ·v) ·αA(bc) + αA(bc) ·αV (v ·a) + αA(bc) ·αV (a ·v) + αV (v · b) ·αA(ca) + αV (b ·v) ·αA(ca) + αA(ca) ·αV (v · b) + αA(ca) ·αV (b ·v) + αV (v · c) ·αA(ab) + αV (c ·v) ·αA(ab) + αA(ab) ·αV (v · c) + αA(ab) ·αV (c ·v) (by a straightforward computation) = {αV (v ·a) ·αA(bc) + αV (v · b) ·αA(ca) + αV (v · c) ·αA(ab)} + {αA(bc) · (αV (v) ·αA(a)) + αA(ca) · (αV (v) ·αA(b)) + αA(ab) · (αV (v) ·αA(c))} + {αA(bc) ·αV (a ·v) + αA(ca) ·αV (b ·v) hom-jordan and hom-alternative bimodules 87 + αA(ab) ·αV (c ·v)} + {(αA(a) ·αV (v)) ·αA(bc) + (αA(b) ·αV (v)) ·αA(ca) + (αA(c) ·αV (v)) ·αA(ab)} (rearranging terms and noting that ρ1 and ρ2 are morphisms) = {(αV (v) · bc) ·α2A(a) + (αV (v) · ca) ·α 2 A(b) + (αV (v) ·ab) ·α 2 A(c)} + {(bc ·αV (v)) ·α2A(a) + (ca ·αV (v)) ·α 2 A(b) + (ab ·αV (v)) ·α 2 A(c)} + {α2A(a) · (bc ·αV (v)) + α 2 A(b) · (ca ·αV (v)) + α 2 A(c) · (ab ·αV (v))} + {α2A(a) · (αV (v) · bc) + α 2 A(b) · (αV (v) · ca) + α 2 A(c) · (αV (v) ·ab)} (by (9), (11) and (21)) = {(αV (v) � bc) ·α2A(a) + (αV (v) � ca) ·α 2 A(b) + (αV (v) �ab) ·α 2 A(c)} + {α2A(a) · (αV (v) � bc) + α 2 A(b) · (αV (v) � ca) + α 2 A(c) · (αV (v) �ab)} (by the definition of �) = (αV (v) � bc) �α2A(a) + (αV (v) � ca) �α 2 A(b) + (αV (v) �ab) �α 2 A(c) (again by the definition of �). Therefore, we get (17). Finally, we have: αV (v �a) �αA(bc) + αV (v � b) �αA(ca) + αV (v � c) �αA(ab) = αV (v ·a) ·αA(bc) + αV (a ·v) ·αA(bc) + αA(bc) ·αV (v ·a) + αA(bc) ·αV (a ·v) + αV (v · b) ·αA(ca) + αV (b ·v) ·αA(ca) + αA(ca) ·αV (v · b) + αA(ca) ·αV (b ·v) + αV (v · c) ·αA(ab) + αV (c ·v) ·αA(ab) + αA(ab) ·αV (v · c) + αA(ab) ·αV (c ·v) (by a straightforward computation) = {αV (v ·a) ·αA(bc) + αV (v · b) ·αA(ca) + αV (v · c) ·αA(ab)} + {(αV (a ·v) ·αA(b)αA(c) + (αV (b ·v)) ·αA(c)αA(a) + (αV (c ·v) ·αA(a)αA(b)} + {αA(bc) ·αV (a ·v) + αA(ca) ·αV (b ·v) + αA(ab) ·αV (c ·v)} + {αA(b)αA(c) ·αV (v ·a) + αA(c)αA(a) ·αV (v · b) + αA(a)αA(b) ·αV (v · c)} (rearranging terms and using the multiplicativity of αA) = {((v ·a) ·αA(b)) ·α2A(c)︸ ︷︷ ︸ 1 + ((v · c) ·αA(b)) ·α2A(a)︸ ︷︷ ︸ 2 + α2V (v) · ((ac)αA(b))︸ ︷︷ ︸ 5 } 88 s. attan, h. hounnon, b. kpamegan + {((a ·v) ·αA(b)) ·α2A(c)︸ ︷︷ ︸ 1 +((a ·v) ·αA(c)) ·α2A(b) + ((b ·v) ·αA(c)) ·α2A(a) + ((b ·v) ·αA(a)) ·α 2 A(c) + ((c ·v) ·αA(a)) ·α2A(b) + ((c ·v) ·αA(b)) ·α 2 A(a)︸ ︷︷ ︸ 2 } + {α2A(c) · (αA(b) · (a ·v))︸ ︷︷ ︸ 3 + α2A(a) · (αA(b) · (c ·v))︸ ︷︷ ︸ 4 + ((ac)αA(b)) ·α2V (v)︸ ︷︷ ︸ 5 } + {α2A(b) · (αA(c) · (v ·a)) + α2A(c) · (αA(b) · (v ·a))︸ ︷︷ ︸ 3 +α2A(a) · (αA(c) · (v · b)) + α2A(c) · (αA(a) · (v · b)) + α 2 A(a) · (αA(b) · (v · c))︸ ︷︷ ︸ 4 + α2A(b) · (αA(a) · (v · c))} (by (10), (12), (14) and (15)) = ((v �a) ·αA(b)) ·α2A(c) + ((v � c) ·αA(b)) ·α 2 A(a) + α2A(c) · (αA(b) · (v �a)) + α 2 A(a) · (αA(b) · (v � c)) + α2V (v) � ((ac)αA(b)) + ((a ·v) ·αA(c)) ·α 2 A(b) + ((b ·v) ·αA(c)) ·α2A(a) + ((b ·v) ·αA(a)) ·α 2 A(c) + ((c ·v) ·αA(a)) ·α2A(b) + α 2 A(b) · (αA(c) · (v ·a)) + α2A(a) · (αA(c) · (v · b)) + α 2 A(c) · (αA(a) · (v · b)) + α2A(b) · (αA(a) · (v · c)) = ((v �a) ·αA(b)) ·α2A(c) + ((v � c) ·αA(b)) ·α 2 A(a) + α2A(c) · (αA(b) · (v �a)) + α 2 A(a) · (αA(b) · (v � c)) + α2V (v) � ((ac)αA(b)) + (αA(a) · (v · c)) ·α 2 A(b) + (αA(b) · (v · c)) ·α2A(a) + (αA(b) · (v ·a)) ·α 2 A(c) + (αA(c) · (v ·a)) ·α2A(b) + α 2 A(b) · ((c ·v) ·αA(a)) + α2A(a) · ((c ·v) ·αA(b)) + α 2 A(c) · ((a ·v) ·αA(b)) + α2A(b) · ((a ·v) ·αA(c)) (by (21)) hom-jordan and hom-alternative bimodules 89 = ((v �a) ·αA(b)) ·α2A(c) + ((v � c) ·αA(b)) ·α 2 A(a) + α2A(c) · (αA(b) · (v �a)) + α 2 A(a) · (αA(b) · (v � c)) + α2V (v) � ((ac)αA(b)) + α 2 A(a) · ((v · c) ·αA(b))︸ ︷︷ ︸ 6 + (αA(b) · (v · c)) ·α2A(a)︸ ︷︷ ︸ 7 + (αA(b) · (v ·a)) ·α2A(c)︸ ︷︷ ︸ 8 + α2A(c) · ((v ·a) ·αA(b))︸ ︷︷ ︸ 9 + (αA(b) · (c ·v)) ·α2A(a)︸ ︷︷ ︸ 7 + α2A(a) · ((c ·v) ·αA(b))︸ ︷︷ ︸ 6 + α2A(c) · ((a ·v) ·αA(b))︸ ︷︷ ︸ 9 + (αA(b) · (a ·v)) ·α2A(c))︸ ︷︷ ︸ 8 (again by (21)) = ((v �a) ·αA(b)) ·α2A(c)︸ ︷︷ ︸ 10 + ((v � c) ·αA(b)) ·α2A(a)︸ ︷︷ ︸ 11 + α2A(c) · (αA(b) · (v �a))︸ ︷︷ ︸ 13 + α2A(a) · (αA(b) · (v � c))︸ ︷︷ ︸ 12 + α2V (v) � ((ac)αA(b)) + α 2 A(a) · ((v � c) ·αA(b))︸ ︷︷ ︸ 12 + (αA(b) · (v � c)) ·α2A(a)︸ ︷︷ ︸ 11 + (αA(b) · (v �a)) ·α2A(c)︸ ︷︷ ︸ 10 + α2A(c) · ((v �a) ·αA(b))︸ ︷︷ ︸ 13 = ((v �a) �αA(b)) ·α2A(c) + ((v � c) �αA(b)) ·α 2 A(a) + α2A(a) · ((v � c) �αA(b)) + α 2 A(c) · ((v �a) �αA(b)) + α2V (v) � ((ac)αA(b)) = ((v �a) �αA(b)) �α2A(c) + ((v � c) �αA(b)) �α 2 A(a) + α2V (v) � ((ac)αA(b)) which is (18). The following result will be used below. It gives a relation between Hom- associative modules and special Hom-Jordan modules. 90 s. attan, h. hounnon, b. kpamegan Lemma 4.11. Let (A,µ,αA) be a Hom-associative algebra and (V,αV ) be a Hom-module. (i) If (V,αV ) is a right Hom-associative A-module with the structure maps ρr then (V,αV ) is a right special Hom-Jordan A +-module with the same structure map ρr. (ii) If (V,αV ) is a left Hom-associative A-module with the structure maps ρl then (V,αV ) is a left special Hom-Jordan A +-module with the same structure map ρl. Proof. It also suffices to prove (14) and (15). (i) If (V,αV ) is a right Hom-associative A-module with the structure map ρr then for all (x,y,v) ∈ A×A×V , αV (v)·(a∗b) = αV (v)·(ab)+αV (v)·(ba) = (v ·a)·αA(b) + (v ·b)·αA(a) where the last equality holds by (5). Then (V,αV ) is a right special Hom-Jordan A+-module. (ii) If (V,αV ) is a left Hom-associative A-module with the structure map ρl then for all (x,y,v) ∈ A×A×V , (a∗b)·αV (v) = (ab)·αV (v)+(ba)·αV (v) = αA(a)·(b·v) +αA(b)·(a·v) where the last equality holds by (4). Then (V,αV ) is a left special Hom-Jordan A+-module. Now, we prove that a Hom-associative module gives rise to a Hom-Jordan module for its plus Hom-algebra. Proposition 4.12. Let (A,µ,αA) be a Hom-associative algebra and (V,ρ1,ρ2,αV ) be a Hom-associative A-bimodule. Then (V,ρl,ρr,αV ) is a Hom-Jordan A+-bimodule where ρl and ρr are defined as in (22). Proof. The proof follows from Lemma 4.11 , the Hom-associativity condi- tion (6) and Theorem 4.10. The following elementary result will be used below. It gives a property of a module Hom-associator. Lemma 4.13. Let (A,µ,αA) be a Hom-Jordan algebra and (V,αV ) be an Hom-Jordan A-bimodule with the structure maps ρl and ρr. Then αnV ◦asA,V ◦ IdA⊗V⊗A = asA,V ◦ (α ⊗n A ⊗α ⊗n V ⊗α ⊗n A ) . (23) hom-jordan and hom-alternative bimodules 91 Proof. Using twice the fact that ρl and ρr are morphisms of Hom-modules, we get αnV ◦asA,V ◦ IdA⊗V⊗A = αnV ◦ (ρr ◦ (ρl ⊗αA) −ρl ◦ (αA ⊗ρr)) = αnV ◦ρr ◦ (ρl ⊗αA) −α n V ◦ρl ◦ (αA ⊗ρr) (linearity of α n V ) = ρr ◦ (αnV ◦ρl ⊗α n+1 A ) −ρl ◦ (α n+1 A ⊗α n V ◦ρr) = ρr ◦ (ρl ◦ (αnA ⊗α n V ) ⊗α n+1 A ) −ρl ◦ (α n+1 A ⊗ρr ◦ (α n V ⊗α n A)) = (ρr ◦ (ρl ⊗αA) −ρl ◦ (αA ⊗ρr)) ◦ (α⊗nA ⊗α ⊗n V ⊗α ⊗n A ) = asA,V ◦ (α⊗nA ⊗α ⊗n V ⊗α ⊗n A ). That ends the proof. The next result is similar to the one of Proposition 3.7, but an additional condition is needed. Proposition 4.14. Let (A,µ,αA) be a Hom-Jordan algebra and (V,αV ) be a Hom-Jordan A-bimodule with the structure maps ρl and ρr. Suppose that there exists n ∈ N such that αnV = IdV . Then the maps ρ (n) l = ρl ◦ (α n A ⊗ IdV ) , (24) ρ(n)r = ρr ◦ (IdV ⊗α n A) (25) give the Hom-module (V,αV ) the structure of a Hom-Jordan A-bimodule that we denote by V (n). Proof. Since the structure map ρl is a morphism of Hom-modules, we get: αV ◦ρ (n) l = αV ◦ρl ◦ (α n A ⊗ IdV ) (by (24)) = ρl ◦ (αn+1A ⊗αV ) = ρl ◦ (αnA ⊗ IdV ) ◦ (αA ⊗αV ) = ρ (n) l ◦ (αA ⊗αV ) Then, ρ (n) l is a morphism. Similarly, we get that ρ (n) r is a morphism and that (16) holds for V (n). Next, we compute 92 s. attan, h. hounnon, b. kpamegan (a,b,c) asA,V (n) (αA(a),αV (v),ab) = (a,b,c){ρ (n) r (ρ (n) l (αA(a),αV (v)),αA(bc))−ρ (n) l (α 2 A(a),ρ (n) r (αV (v),bc))} = (a,b,c){ρr(ρ (n) l (αA(a),αV (v)),α n+1 A (bc))−ρl(α n+2 A (a),ρ (n) r (αV (v),bc))} = (a,b,c){ρr(ρl(α n+1 A (a),αV (v)),α n+1 A (bc))−ρl(α n+2 A (a),ρr(αV (v),α n A(bc))} = (a,b,c){ρr(ρl(α n+1 A (a),αV (v)),αA(α n A(bc))) −ρl(αA(αn+1A (a)),ρr(αV (v),α n A(bc))} = (a,b,c) asA,V (α n+1 A (a),αV (v),α n A(bc)) = (a,b,c) asA,V (α n+1 A (a),α n+1 V (v),α n A(bc)) (by the hypothesis αV = α n+1 V ) = αnV ( (a,b,c) asA,V (αA(a),αV (v),bc)) (by (23) and the linearity of α n V ) = 0 (by (19) in V ). Then we get (19) for V (n). Finally remarking that asA,V (n) (ρ n r (v,a),αA(b),αA(c)) = asA,V (n) (v ·α n A(a),αA(b),αA(c)) = ρnr (ρ n r (v ·α n A(a),αA(b),α 2 A(c)) −ρ n r (αV (v) ·α n+1 A (a),µ(αA(b),α(A(c)) = ρr(ρr(v ·αnA(a),α n+1 A (b),α n+2 A (c)) −ρr(αV (v) ·αn+1(a),µ(αn+1A (b),α n+1 A (c)) = αA,V (v ·αnA(a),α n+1 A (b),α n+1 A (c)) , and similarly asA,V (n) (ρ n r (v,c),αA(b),αA(a)) = asA,V (v ·α n A(c),α n+1 A (b),α n+1 A (a)) , asA,V (n) (ac,αA(b),αV (v)) = asA,V (α n A(a)α n A(c),α n+1 A (b),αV (v)) (20) is proved for V (n) as it follows: asA,V (n) (ρ n r (v,a),αA(b),αA(c)) + asA,V (n) (ρ n r (v,c),αA(b),αA(a)) + asA,V (n) (ac,αA(b),αV (v)) = αV (v ·αnA(a),α n+1 A (b),α n+1 A (c)) + asA,V (v ·α n A(c),α n+1 A (b),α n+1 A (a)) + asA,V (α n A(a)α n A(c),α n+1 A (b),αV (v)) hom-jordan and hom-alternative bimodules 93 = αV (v ·αnA(a),αA(α n A(b)),αA(α n A(c))) + asA,V (v ·αnA(c),αA(α n A(b)),αA(α n A(a))) + asA,V (α n A(a)α n A(c),αA(α n A(b)),αV (v)) = 0(by (20) in V ). We conclude that V (n) is a Hom-Jordan A-bimodule. Example 4.15. Consider the Hom-Jordan algebra A+ of the Examples 2.10 and the subspace V = span(e1,e3) of A. Then (V,µV ,αV ) is a Hom- ideal of A+ where µV = µA|V and αV = αA|V . It follows that (V,ρl,ρr,αV ) is a Hom-Jordan A+-bimodule where ρl and ρr are defined as in Examples 4.7. We have α2V = IdV , then by Proposition 4.14, the structure maps ρ (2) l = ρl ◦ (α2A ⊗IdV ) and ρ (2) r = ρr ◦ (IdV ⊗α2A) give the Hom-module (V,αV ) the structure of a Hom-Jordan A+-bimodule that we denote by V (2). Corollary 4.16. Let (A,µ,αA) be a Hom-Jordan algebra and (V,αV ) be a Hom-Jordan A-bimodule with the structure maps ρl and ρr such that αV is an involution. Then (V,αV ) is a Hom-Jordan A-bimodule with the structure maps ρ (2) l = ρl ◦ (α 2 A ⊗ IdV ) and ρ (2) r = ρr ◦ (IdV ⊗α2A). Example 4.17. Consider the Hom-Jordan algebra B+ of the Examples 2.10 and the subspace V = span(e1,e2) of B. Then (V,µV ,αV ) is a Hom-ideal of B+ where µV = µB|V and αV = αB|V . Therefore (V,ρl,ρr,αV ) is a Hom- Jordan B+-bimodule where ρl and ρr are defined as in Examples 4.7. Note that αV is involutive, i.e., α 2 V = IdV , then by Corollary 4.16, the structure maps ρ (2) l = ρl ◦(α 2 B ⊗IdV ) and ρ (2) r = ρr ◦(IdV ⊗α2B) give the Hom-module (V,αV ) the structure of a Hom-Jordan B+-bimodule. The following result is similar to theorem 3.8. It says that Jordan bimod- ules can be deformed into Hom-Jordan bimodules via an endomorphism. Theorem 4.18. Let (A,µ) be a Jordan algebra, V be a Jordan A-bimod- ule with the structure maps ρl and ρr, αA be an endomorphism of the Jordan algebra A and αV be a linear self-map of V such that αV ◦ρl = ρl◦(αA⊗αV ) and αV ◦ ρr = ρr ◦ (αV ⊗ αA). Write AαA for the Hom-Jordan algebra (A,µαA,αA) and VαV for the Hom-module (V,αV ). Then the maps: ρ̃l = αV ◦ρl and ρ̃r = αV ◦ρr give the Hom-module VαV the structure of a Hom-Jordan AαA-bimodule. 94 s. attan, h. hounnon, b. kpamegan Proof. It is easy to prove that the relation (16) for VαV holds and both maps ρ̃l, ρ̃r are morphisms. Remarking that asA,VαV = α 2 V ◦asA,V (26) we first compute (a,b,c) asA,VαV (αA(a),αV (v),µαA(b,c)) = (a,b,c) α 2 V (asA,V (αA(a),αV (v),αA(bc))) (by (26)) = (a,b,c) α 3 V ((asA,V (a,v,bc)) (by (23)) = α3V ( (a,b,c) (asA,V (a,v,bc)) = 0 (by (19) in V ) and then, we get (19) for VαV . Finally, we get asA,VαV (ρ̃r(v,a),αA(b),αA(c)) + asA,VαV (ρ̃r(v,c),αA(b),αA(a)) + asA,VαV (µαA(a,c),αA(b),αV (v)) = α2V (asA,V (ρ̃r(v,a),αA(b),αA(c))) + α 2 V (asA,V (ρ̃r(v,c),αA(b),αA(a))) + α2V (asA,V (µαA(a,c),αA(b),αV (v))) (by (26)) = α2V (asA,V (αV (v ·a),αA(b),αA(c))) + α2V (asA,V (αV (v · c),αA(b),αA(a))) + α2V (asA,V (αA(ac),αA(b),αV (v))) = α3V (asA,V (v ·a,b,c)) + α 3 V (asA,V (v · c,b,a)) + α3V (asA,V (ac,b,v)) (by 23) = α3V (asA,V (v ·a,b,c) + asA,V (v · c,b,a) + asA,V (ac,b,v)) = 0 (by (20) in V ) which is (20) for VαV . Therefore the Hom-module VαV has a Hom-Jordan AαA-bimodule structure. Corollary 4.19. Let (A,µ) be a Jordan algebra, V be a Jordan A- bimodule with the structure maps ρl and ρr, αA be an endomorphism of the Jordan algebra A and αV be a linear self-map of V such that αV ◦ρl = ρl ◦ (αA ⊗αV ) and αV ◦ρr = ρr ◦ (αV ⊗αA). Moreover, suppose that there exists n ∈ N such that αnV = IdV . Write AαA for the Hom-Jordan algebra (A,µαA,αA) and VαV for the Hom-module hom-jordan and hom-alternative bimodules 95 (V,αV ). Then the maps: ρ̃ (n) l = ρl ◦ (α n+1 A ⊗αV ) and ρ̃ (n) r = ρr ◦ (αV ⊗α n+1 A ) (27) give the Hom-module Vα the structure of a Hom-Jordan AαA-bimodule for each n ∈ N. Proof. The proof follows from Proposition 4.14 and Theorem 4.18. Similarly to Hom-alternative algebras, the split null extension, determined by the given bimodule over a Hom-Jordan algebra, is constructed as follows: Theorem 4.20. Let (A,µ,αA) be a Hom-Jordan algebra and (V,αV ) be a Hom-Jordan A-bimodule with the structure maps ρl and ρr. Then (A⊕V,µ̃, α̃) is a Hom-Jordan algebra where µ̃ : (A⊕V )⊗2 → A⊕V , µ̃(a+m,b+n) := ab+a·n+m·b and α̃ : A⊕V → A⊕V , α̃(a + m) := αA(a) + αV (m) Proof. First, the commutativity of µ̃ follows from the one of µ. Next, the multiplicativity of α̃ with respect to µ̃ follows from the one of α with respect to µ and the fact that ρl and ρr are morphisms of Hom-modules. Finally, we prove the Hom-Jordan identity (3) for E = A⊕V as it follows asE(µ̃(x + m,x + m), α̃(y + n), α̃(x + m)) = µ̃(µ̃(µ̃(x + m,x + m), α̃(y + n)), α̃2(x + m)) − µ̃(α̃(µ̃(x + m,x + m)), µ̃(α̃(y + n), α̃(x + m))) = µ̃(µ̃(x2 + x ·m + m ·x,αA(y) − µ̃(αA(x2) + αV (n)),α2A(x) + α 2 V (m)) + αV (x ·m) + αV (m ·x), µ̃(αA(y) + αV (n),αA(x) + αV (m))) = µ̃(x2αA(y) + x 2 ·αV (n) + (x ·m) ·αA(y) + (m ·x) ·αA(y),α2A(x) + α2V (m)) − µ̃(α 2 A(x 2) + αV (x ·m) + αV (m ·x),αA(y)αA(x) + αA(y) ·αV (m) + αV (n) ·αA(x)) = (x2αA(y))α 2 A(x) + (x 2αA(y)) ·α2V (m) + (x 2 ·αV (n)) ·α2A(x) + ((x ·m) ·αA(y)) ·α2A(x) + ((m ·x) ·αA(y)) ·α 2 A(x)) −αA(x2)(αA(y)αA(x)) −αA(x2) · (αA(y) ·αV (m)) −αA(x2) · (αV (n) ·αA(x)) −αV (x ·m) · (αA(y)αA(x)) −αV (m ·x) · (αA(y)αA(x)) 96 s. attan, h. hounnon, b. kpamegan = asA(x 2,αA(y),αA(x)) + asA,V (x 2,αA(y),αV (m)) + asA,V (x 2,αV (n),αA(x)) + asA,V (x ·m,αA(y),αA(x)) + asA,V (m ·x,αA(y),αA(x)) = asA,V (m ·x,αA(y),αA(x)) + asA,V (m ·x,αA(y),αA(x))︸ ︷︷ ︸ 0 + asA,V (x 2,αV (n),αA(x)) + asA,V (x 2,αA(y),αV (m))︸ ︷︷ ︸ 0 + asA(x 2,αA(y),αA(x))︸ ︷︷ ︸ 0 = 0 , where the first 0 follows from (20), the second from (19) (see Remarks 4.6) and the last from the Hom-Jordan identity (3) in A. We conclude then that (A⊕V,µ̃, α̃) is a Hom-Jordan algebra. Similarly as Hom-alternative algebra case, let give the following: Remark 4.21. Consider the split null extension A⊕V determined by the Hom-Jordan bimodule (V,αV ) for the Hom-Jordan algebra (A,µ,αA) in the previous theorem. Write elements a + v of A⊕V as (a,v). Then there is an injective homomorphism of Hom-modules i : V → A⊕V given by i(v) = (0,v) and a surjective homomorphism of Hom-modules π : A ⊕ V → A given by π(a,v) = a. Moreover, i(V ) is a Hom-ideal of A⊕V such that A⊕V/i(V ) ∼= A. On the other hand, there is a morphism of Hom-algebras σ : A → A⊕V given by σ(a) = (a, 0) which is clearly a section of π. Hence, we obtain the abelian split exact sequence of Hom-Jordan algebras and (V,αV ) is a Hom-Jordan bimodule for A via π. References [1] H. Ataguema, A. Makhlouf, S.D. Silvestrov, Generalization of n- ary Nambu algebras and beyond, J. Math. Phys. 50 (8) (2009), 083501, 15 pp. [2] I. Bakayoko, B. 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Makhlouf, Structure and classification of Hom-associative algebras. arXiv:1906.04969V1[math.RA] arXiv:0712.3515v1 arXiv:0812.4695v1 arXiv:1906.04969V1[math.RA] Introduction Preliminaries Hom-alternative bimodules Hom-Jordan bimodules